BBC (British Broadcasting Corporation)
(launched as a radio network in 1922, and as a TV network in 1932)
IMDb; official website; TV Tropes; Wikia; Wikipedia

I've never actually seen anything directly on the BBC (or any other British networks), of course. But I've seen numerous BBC shows on American channels like PBS and BBC America, as well as Canadian channel CBC. Some things I first saw on VHS, and much later, there were things I watched online. Also I should say, "BBC" is more than just one channel, but I'm going to list all the BBC shows I've seen on a single page, anyway, just under different headings for the specific channels. It should be noted that many of these shows may have aired on various BBC channels in the UK, particularly in reruns, but I'll try to list the original channel for each show. (I may sometimes get it wrong; certainly the BBC website itself is no help, as it tends to list things only on whichever channel aired them most recently.)

BBC One
Doctor Who (11/23/1963 - 12/6/1989)
Monty Python's Flying Circus (10/5/1969 - 1/18/1973; s4 aired on BBC2)
Are You Being Served? (9/8/1972 - 4/1/1985)
Blake's 7 (1/2/1978 - 12/21/1981)
K-9 & Company (12/28/1981; special)
Blackadder (6/15/1983 - 11/2/1989)
The Five Doctors (11/25/1983; special; aired two days earlier on PBS)
The Chronicles of Narnia (11/13/1988 - 12/23/1990; serials)
Blackadder's Christmas Carol (12/23/1988; special)
May to December (4/2/1989 - 5/27/1997)
Absolutely Fabulous (11/12/1992 - 5/11/1995; followed by a 2-part special in 1996, 2 more series in 2001-2003, more specials; and a theatrical film in 2016)
The Buccaneers (2/5/1995 - 3/5/1995; miniseries)
Doctor Who (5/27/1996; TV movie; FOX, 5/14/1996)
Doctor Who: The Curse of the Fatal Death (3/12/1999; special)
Gormenghast (1/17/2000 - 2/7/2000; serial)
My Hero (2/4/2000 - 9/10/2006)
My Family (9/19/2000 - 9/2/2011; I didn't see much of this)
Spooks (5/13/2002 - 10/23/2011; aired in the US as "MI-5")
Carrie's War (1/1/2004; TV movie)
Doctor Who (3/26/2005 - present)
My Family and Other Animals (12/27/2005; TV movie)
Robin Hood (10/7/2006 - 6/27/2009)
The Ruby in the Smoke (12/27/2006; TV movie)
Dracula (12/28/2006; TV movie)
The Sarah Jane Adventures: Invasion of the Bane (1/1/2007; special)
Doctor Who: The Infinite Quest (4/2/2007 - 6/29/2007; microseries)
Jekyll (6/16/2007 - 7/28/2007; serial)
The Sarah Jane Adventures (9/24/2007 - 10/18/2011; also aired on CBBC)
The Sarah Jane Adventures: From Raxacoricofallapatorius with Love (3/13/2009; special)
Torchwood (series 3-4) (7/6/2009 - 9/9/2011; series 1 aired on BBC3 and series 2 aired on BBC2)
Sherlock (7/25/2010 - 1/15/2017)
Sherlock: The Abominable Bride (1/1/2016; special)
Ordeal by Innocence (4/1/2018 - 4/15/2018; serial)
His Dark Materials (11/3/2019 - present; airs on HBO in the US)
A Christmas Carol (12/22-24/2019; miniseries; aired on FX in the US)

BBC Two
Monty Python's Flying Circus (10/31/1974 - 12/5/1974; s1-3 aired on BBC1)
Fawlty Towers (9/19/1975 - 10/25/1979)
The Hitchhiker's Guide to the Galaxy (1/5/1981 - 2/9/1981; serial)
The Young Ones (11/9/1982 - 6/19/1984)
Yes, Prime Minister (1/9/1986 - 1/28/1988)
Red Dwarf (2/15/1988 - 4/5/1999; revival series on Dave, which I haven't seen, 2009-present)
Hippies (11/12/1999 - 12/17/1999)
Look Around You (10/10/2002 - 3/7/2005)
Torchwood (series 2) (1/16/2008 - 4/4/2008)
Krod Mandoon and the Flaming Sword of Fire (6/11/2009 - 7/6/2009)
The Hollow Crown (6/30/2012 - 7/21/2012; TV movies)
The Wrong Mans (9/24/2013 - 12/23/2014; Hulu)
An Adventure in Space and Time (11/21/2013; TV movie)

BBC Three
The Mighty Boosh (5/18/2004 - 12/20/2007)
Torchwood (series 1) (10/22/2006 - 1/1/2007)
Being Human (2/18/2008 - 3/10/2013)
Class (10/22/2016 - 12/3/2016)

BBC Four

CBBC
The Sarah Jane Adventures (9/24/2007 - 10/18/2011; also aired on BBC 1)
Wolfblood (9/10/2012 - 5/1/2017)

BBC Red Button
Doctor Who: Dreamland (11/21/2009 - 11/26/2009; microseries)