I'm quite a fan so I'll mention a few - since they are all free on iTunes I won't give links, but the non-BBC ones tend to have easily accessed websites. Remember all of these have an archive so you can start from the beginning.
BBC.
Moral Maze
Great Lives
The Why Factor
Beyond Belief
Thinking Allowed
In Our Time (with archives on Science, Philosophy, History, Culture)
The History of the World in 100 Objects.
Soul Music (which focuses on a song or piece of music and asks people why the piece is important to then - some, such as A Change is Gonna Come, are very moving).
Wireless Nights with Jarvis Cocker, looking at what people get up to at night.
The History of Rome and Revolutions - both by Mike Duncan. The first starts slowly, by his own admission, but then finds its stride: just 170+ episodes to listen to. The latter covers the various revolutions in more modern times starting with the 17th century civil war here, American Independence, the French Revolution and so on - currently on 1848 events in Europe.
Attaboy Clarence - for anyone interested in the heyday of Hollywood - includes ads from the time and also radio-plays, some of which are abridged versions of well known films with the original cast doing the radio version - for example Cagney doing Angels with Dirty Faces for radio.
The War - episodes of radio shows compiled for US troops abroad during WW2: lots of music from the time plus comedy (Bob Hope etc) - the flagrant advertising (especially cigarettes) sound quite odd now.
For those who like fantasy try Drabblecast.
For journalism try The Guardian Long Reads
Best of Old Time Radio - US radio shows from the pre-TV era mixing comedy (Jack Benny etc and early radio sit-coms) with drama - for example episodes of Dragnet, Gunsmoke etc before they migrated to TV. Again the mid-show ads are hilarious.
The Mortified Podcast - adults reading their old teenage diaries to an audience: mostly US but can be quite funny.
