I listen to podcasts a fair bit either while running or on my commute:
The Dollop - Two idiot comedians cover weird bits of American history.
Hardcore History - Former talk radio presenter and history grad does really engaging pieces on whatever areas of history interest him. Makes for great overviews.
12 Byzantine Rulers - By the guy who started the whole history podcasting thing
Talking History - The Unification of Italy
The China History Podcast - A very laidback Californian and long-term China hand presents Chinese history in a very approachable manner
Popup Chinese - Bitesize Chinese lessons
Sinica - The best English language podcast out there on current affairs, culture etc in China. Covers a variety of stuff like "shanzhai" counterfeit culture, Chinese attitudes to air pollution, golf, censorship, Chinese cruise ship tourism, birdwatching and conservation in Beijing, women's rights, neo-Maoists etc.
In Our Time - A BBC radio programme - 3 or 4 professors are invited each week to talk on a specific subject of theirs for the layperson. A bit rushed along by the presenter to fit the 40 minute format but it never meanders.
Kermode & Mayo Film Reviews - A couple of wittering, middle-aged fuddy duddies talk about cinema. A BBC radio programme with a slightly cultish following in the UK.
FilmSpotting
Serial - Season 1 covered a murder conviction case which may have been a miscarriage of justice. Season 2 covers the story of Bo Bergdahl in depth.
Radiolab - science & philosophy-ish
99% Invisible - town planning, design, architecture, technology, infrastructure and a smidgin of history
Africa Today - Weekly African news roundup from the BBC
BBC World Service Documentary podcasts