Following a worldwide search, I can confidently assert that the BBC’s In Our Time is the best podcast in English. Its remit in the history if ideas - science, philosophy, history, religion. What gives In Our Time its edge is that it’s broadcast live (9am Thursdays) without editing. There’s a newsletter and a listeners debate page, sometimes very illuminating.
You can access a substantial archive of past episodes, but you can't download, you can only “listen again” online. Downloading must be done within 7 days of the episode being broadcast. This rule applies to all the BBC’s output so far as I know, but doesn’t seem to apply to other broadcasters.
I often make notes of In Our Time episodes, including extracts from listeners’ comments and the newsletter. Available on request.
My best 3 - The Translation Movement, The History of Optics, The Nicene Creed
Other BBC favourites are The Forum (World Service), Start the Week, and Woman’s Hour, also two programmes dedicated to religion: Sunday, and Beyond Belief.
Next a few words about science podcasts. There’s a strong field, and I like that all of them touch on the philosophy of science, including science and religion issues. Top of my list is The Guardian’s Science Weekly, with the Nature podcast next.
The SETI institute’s Are We Alone needs a mention. You may wonder if this is a restricted zone for ET enthusiasts. Not so! It's good science, often lacking any obvious connection with the Mekon. If the presenter Seth Shostak ever tires of physics he could make a living in comedy.
Also worth a look : Science and the City (New York Academy of Sciences), and the CBC’s Quirks and Quarks
That’s all I want to say about science.
There's a separate page for past series of which my best 3 are: Icons of Irish Science, Judging Dev, and Our War (Ireland and the First World War)
I also need to mention:-
Words at Large (books), and Ideas (politics and philosophy) are top class. I've already mentioned the CBC science programme Quirks and Quarks,
In 2008 or thereabouts the CBC produced a 24-part series called "How to Think About Science" Top class. My favourite Best of Ideas episode: Living on Oxford Time, 21/7/08. It includes Roger Penrose and his notion that time may not exist, it's all bound up with consciousness.
Top here is The Diane Rehm Show (daily, politics, current affairs), and Fresh Air (extended interviews on almost anything. I sample the site from time to time rather than subscribing to the podcast)
NPR Shuffle is a daily sampler from various mainly current affairs broadcasts. Also try It’s All Politics (weekly, takes a narrow view of “politics”, but informed and entertaining) and The Talk of the Nation Opinion Page (daily interview and phone-in with the writer of a controversial op-ed). NPR also has a science programme called Radiolab which I haven’t heard but it has won an award.
1. Background Briefing. Episodes worth mention :
2008-03-30 The three trillion dollar war - Joseph Stilglitz on the Iraq war
2008-04-06 The climate engineers - the politics and ethics of global climate control is even more interesting than the science. Transcripts available.
2. Late Night Live – daily - politics, history, philosophy, current affairs, sometimes excellent, sometimes annoying.
3. The Ark is no longer broadcast but the archive is still available on the website. History of religion, taken in a broad sense - for example one episode was devoted to the claim that an Arab astronomer anticipated Copernicus by 200 years. This little programme used to pack a lot into 20-minute episodes. Transcripts available.4. The Philosophers Zone and All in the Mind.