PBS (Public Broadcasting Service; see also PBS Kids)
(launched in 1970)
IMDb; official website; TV Tropes; Wikipedia

First of all, it can be hard sometimes to know whether something you're watching is actually broadcast nationally by PBS, or just locally by one's own public television station that happens to be affiliated with PBS. Secondly, a lot of the stuff I've watched on my local channel (whether it was a local or national broadcast) is actually acquired from the UK. So I'll try to limit myself to listing programs that I'm sure are actual PBS shows. To save some time, though, I do have a subsection of TV reviews that is specifically for anything I've seen on Mystery! and Masterpiece Theatre. (Those are, of course, PBS anthology series that mostly showcase British serials and TV movies.)

Captain Kangaroo (10/3/1955 - 12/8/1984; originally on CBS, but I only remember watching it on PBS. I guess those were reruns.)
Mister Rogers' Neighborhood (2/19/1968 - 8/31/2001)
Sesame Street (10/10/1969 - present; also aired on HBO from 2016-2020, and since then on HBO Max)
The Electric Company (10/25/1971 - 4/15/1977)
WIll Rogers' USA (?/?/1972; TV play)
The Letter People (1972-?)
Great Performances (11/4/1972 - present; I haven't seen much of this)
Sneak Previews (9/4/1975 - ?/?/1996; I probably didn't see much of this)
Jack Horkheimer: Star Hustler (11/6/1976 - ?; I didn't see much of this)
3-2-1 Contact (1/14/1980 - 11/18/1988)
Reading Rainbow (6/6/1983 - 11/10/2006)
WonderWorks (1984-1992)
French in Action (1987; I haven't seen much of this)
Square One TV (1/26/1987 - 11/6/1992)
The Diaries of Adam and Eve (4/26/1989; TV play)
The Red Green Show (1/4/1991 - 4/7/2006; also on CBC)
Into the Woods (3/15/1991; TV play)
Where in the World is Carmen Sandiego? (9/30/1991 - 12/22/1995)
Ghostwriter (10/4/1992 - 2/12/1995)
Storytime (10/11/1992 - 9/1/1997; not sure how much of this I saw)
I'll Fly Away: Then and Now (10/11/1993; TV movie finale; I'll Fly Away aired on NBC)
The Steven Banks Show (?-?/1994)
The Big Comfy Couch (1/9/1995 - 2/?/2007; originally aired on YTV in Canada)
The 1900 House (6/12/2000 - 7/3/2000; originally on Channel 4 in the UK)
Sagwa, the Chinese Siamese Cat (9/3/2001 - 10/5/2002; I didn't see a lot of this)
Coyote Waits (11/16/2003; TV movie)
A Thief of Time (7/11/2004; TV movie)
Odd Squad (11/26/2014 - present; I've seen maybe a few seasons)
Film School Shorts (9/15/2015 - ?; I haven't seen much of this)

stuff I want to see
Cosmos: A Personal Voyage
The Lathe of Heaven (TV movie)
Powerhouse (I watched this as a little kid, and I'd like to see it again to refresh my memory)
Skinwalkers (TV movie)