Elementary, on CBS
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Caution: spoilers.

Season One
The series is about Sherlock Holmes (Jonny Lee Miller), but it's set in the present, which means it can't help but be compared to the BBC series Sherlock. I like the British show better, but this is decent, too. Anyway, there are some differences from "Sherlock," such as Holmes having recently moved from London to New York City. He's now a consultant for the NYPD, having previously worked with Captain Tommy Gregson (Aidan Quinn), when the American cop had done some work with Scotland Yard some time ago, I guess. (Some websites, including CBS.com, say his name is Tobias, but it's not.) So, Gregson knows how good Sherlock is, even if the amateur detective's demeanor is off-putting to most people he meets. This includes Marcus Bell, a detective in the NYPD who works closely with Gregson. He doesn't like Sherlock at first, but they do eventually come to respect each other.

Meanwhile, Sherlock is recovering from a drug addiction, a fact he is keeping secret from most people, including Gregson. And I guess he doesn't accept money for his consultations, so he gets all his money from his rich and powerful father (whom he detests), who has also hired a New York woman named Joan Watson (Lucy Liu), a former surgeon who quit practicing medicine because of a tragic incident with one of her patients. She's now a "sober companion," which means she lives with and accompanies her client wherever he or she may go. At the moment, this means Sherlock (whose home, which his father owns, is called "The Brownstone"). Of course, Joan reports to Sherlock's father, who would stop supporting Sherlock if he slips back into addiction. Anyway, Watson soon begins to take an interest in helping Sherlock investigate the cases on which he's consulting. Eventually, she ceases to be his sober companion, and instead becomes his protégé, learning to become a detective, herself. So Sherlock gets a new sponsor, named Alfredo Llamosa, whom we see occasionally. And in one episode, we meet an old friend of Sherlock's named Ms. Hudson, who stays with him and Joan briefly. I think we only see her in one other episode (in season 2), but I'm fairly sure she's at least mentioned several other times.

Of course, most episodes are fairly, you know, episodic. The cases are reasonably interesting, though the important thing about the show is the characters. We eventually learn that Sherlock was once in love with a woman named Irene Adler, who was murdered. That's what prompted Sherlock's drug problem. And we learn that if he ever finds whoever killed Irene, he intends to kill him. Which of course does not sit well with Gregson, to say the least. However, Sherlock eventually learns that Irene was killed on the orders of a mysterious criminal mastermind named Moriarty, who becomes Sherlock's nemesis. We don't see Moriarty until the end of the first season, and that... involves a shocking revelation that I won't spoil. But I will say that Moriarty is finally arrested.

Season Two
At the start of season 2, we meet Sherlock's former partner, Gareth Lestrade (Sean Pertwee). He'll show up later in the season as a rival detective. We also meet Sherlock's brother, Mycroft Holmes (Rhys Ifans), who is estranged from Sherlock, but who becomes quite friendly with Joan. And Sherlock and Joan occasionally get help from a cyber-activist group called "Everyone." And at one point, Gregson separates from his wife, Cheryl. (They'll later divorce.) Partway through the season, Detective Bell gets shot, for which he blames Sherlock. (Not that Sherlock did it, just that it was his fault it happened.) The incident leads to Bell temporarily getting a different job, but he eventually returns to his old job. Still, it takes awhile before he forgives Sherlock. Also this season, Sherlock begins sponsoring a recovering addict named Randy. The season ends with a multiple-episode arc involving Mycroft, who renews his relationship with Joan, and encourages her to move out of the Brownstone, and into her own place. Also, some surprising facts about Mycroft are revealed, which I don't want to spoil. But it all leads to a possible new job for Sherlock....

Season Three
At the end of season 2, Sherlock moved back to England to take a job with MI6. At the start of season 3 (several months later), he returns to New York to resume his old life as a consulting detective, after being fired from MI6. Watson is now working as both a private detective and a consultant for the NYPD, but she'll return to frequently working with Sherlock, while also continuing to work on her own cases. Also, she's now dating a man named Andrew (whose surname is given on some sites as Mittal and others as Paek; I have no recollection of hearing it, so I don't know which is right). Meanwhile, Sherlock has brought with him a new protégé, Kitty Winter (Ophelia Lovibond). Back in England, she had been held captive for some time by a serial rapist, and her work with Sherlock was helping her move past that experience. Eventually a case they're working leads to the realization that her former tormentor is now in New York, and doing the same thing to other women. Kitty finally finds him and takes a measure of revenge, then returns to England. Sometime after that, Joan's relationship with Andrew ends in a way that I don't want to spoil, and sometime after that, Joan moves back into the Brownstone. Later in the season, Sherlock decides he doesn't want Alfredo to be his sponsor anymore, so that they can be friends, instead. In the season finale, Alfredo is abducted by Oscar Rankin, a drug addict who had been seen earlier in the season, who had known Sherlock when he was still using. This leads to... something bad, but I'm not positive exactly what. I'll have to wait for season four to find out for sure.

Season Four
So... the final scene of season 3 made me suspect Sherlock had used heroin again, but I felt there was some ambiguity about it. (Maybe it's just me; anyone else watching might have felt certain of it.) Anyway, this season we know for sure that he did. But he's sober again, and eager to get back to working cases. However, he and Joan are no longer allowed to work with the NYPD, due to Sherlock having beaten Oscar Rankin nearly to death last season. But then his father, Morland Holmes (John Noble), comes to New York, and offers to use his influence to get Sherlock and Joan reinstated. Sherlock refuses at first, but eventually agrees. Unbeknownst to him, Joan meets with Morland and warns him not to hurt Sherlock. (Morland's business practices are often unethical and/or illegal, and Joan knows at least enough about it to threaten him.) Meanwhile, Morland decides to stay in New York for some time, and try to reconcile with his son. Also this season, Gregson begins dating a woman named Paige Cowen (Virginia Madsen), who has multiple sclerosis. Also, during one of this season's cases, Sherlock meets a neuro-atypical software engineer named Fiona Helbron, whom he later begins dating. We eventually learn that some time ago, there had been an attempt on Morland's life, in which Sabine, the woman he loved, was killed. He wants to find out who was responsible for that, and Sherlock helps. They eventually learn that the person behind the assassination attempt and Sabine's death was tied to Moriarty's organization. When Moriarty was arrested at the end of season one, Sherlock had assumed the organization would dissolve, but he was wrong. It's currently being run by a professor named Joshua Vikner (Tony Curran). We finally learn the reason for the attempt on Morland's life, and that ultimately leads to a major plot twist, which I won't reveal for now. (But I may have to someday, if a future plotline demands it.) Anyway, that's all I want to say about this season, I guess.

Season Five
This season, a former patient of Joan's named Shinwell Johnson is introduced. He's recently gotten out of prison, and at first I thought it looked like Joan would help with his transitioning back into society. But he eventually becomes a police informant, rejoining his old gang, SBK, in the hopes of taking them down from the inside. So Joan and Sherlock give him some training. Meanwhile, Marcus begins dating a lawyer named Chantal Milner, whose ex-husband, Roy Booker, later makes trouble for them. And Sherlock and Fiona break up. And Kitty returns for a couple of episodes. And eventually, Joan and Sherlock come to a disagreement concerning Shinwell, which leads to some strife between them, though that remains mostly below the surface of their partnership until the end of the season. Shinwell is finally killed by SBK, so Joan and Sherlock overcome their differences and work together to try to finish Shinwell's efforts to dismantle the gang. Meanwhile, the gang's ostensible leader, Bonzi Folsom is killed, which the detectives believe was done by his half brother, Tyus Wilcox, a seemingly legit businessman whom they believe was the gang's real leader. However, the murder apparently emboldens a rival gang, so everyone has to try to forestall an all-out gang war. But Joan's anger at Sherlock again resurfaces, for reasons I don't want to spoil. But unbeknownst to Joan, Sherlock has apparently been experiencing some memory loss and hallucinations, which is likely to be a major factor of next season's storyline....

Season Six
This season aired in spring and summer.
Sherlock learns he has Post-Concussion Syndrome, which may interfere with his ability to continue his work. Meanwhile, he meets a fellow recovering addict named Michael, who says that something Sherlock had said at a support group meeting helped him with his own recovery. Now he wants to help Sherlock, by asking Sherlock to help him. He wants Sherlock to try to find a friend who has gone missing, and he believes keeping his mind busy in that way could help Sherlock overcome his PCS. However, it eventually turns out that Michael is actually a serial killer, though Sherlock can't prove it. Michael leaves NYC for awhile, but says he'll return when Sherlock's PCS has improved. Also this season, Joan considers adoption. And Marcus gets an offer to join the US Marshal Service, which he'll work on preparing for throughout the remainder of the season. And... at the end of season 4, Morland had decided to take over Moriarty's criminal organization, and begin dismantling it from within. This season, there is an attempt on his life, which Sherlock discovers is related to Moriarty having escaped from the FBI. In the penultimate episode of the season, Michael returns, so Sherlock and Joan resume trying to prove his guilt. Michael is also investigated FBI Special Agent Mallick (Parminder Nagra). But Michael ends up being murdered. I don't want to spoil any details of the season finale, but... it ends with Sherlock and Joan relocating to London.

After the season ended, I read in AV Club's review of the season finale that this was originally expected to be the series finale, which I hadn't been aware of, but I had suspected. Because it really seemed like a good way to end the series. However, I also learned in that review that the series had been renewed for another season. So that should be interesting.

Season Seven
This season also aired in spring and summer.
So... I'm still not going to bother revealing why Sherlock and Joan moved to London, but I will say that this season, we ended up seeing them there for just one episode. They had already been working with Scotland Yard for some time, and I really wish we could have seen more of that. (We also got to see Kitty again in that episode.) But at the end of the season premiere, Marcus called to let them know that Captain Gregson had been shot, and was in critical condition. So, the second episode sees Sherlock and Joan returning to New York. (And not long after that, Sherlock finds a way to deal with the problem that had forced them to leave, last season.) While Gregson is in the hospital, his precinct gets a temporary captain named Dwyer, who does double duty overseeing his own precinct as well as Gregson's. But eventually Gregson returns to work. Meanwhile, Sherlock and Joan eventually learn that Gregson's having been shot was tied up in something much larger. It turns out that the head of an information technology company, Odin Reichenbach (James Frain), had developed a system for spying on the online information of people who use his products, in order to predict who is likely to commit serious crimes. He then has those people killed before they can kill others. And he wants Sherlock and Joan to help him refine his determination of who deserves to be killed. Of course, they want no part of it, and he soon becomes an enemy, one with powerful allies. Eventually, Sherlock's father tries to help them against Reichenbach. But it is ultimately Sherlock and Joan who finally put an end to Reichenbach's crimes. I certainly don't intend to spoil how they go about stopping him, in the penultimate episode of the series. But I will say the series finale involves a three-year time jump, and later another one-year time jump. And... everything gets wrapped up fairly neatly, I suppose. And I'd say I'm about equally as happy with this being the series finale as I would have been with last season's finale being the end of the series.