In our time the BBC radio/podcast is very good. Its a panel based show with the host and three experts on the subject. Since its a radio show its tightly run and goes at pace. It covers a very broad set of topics from scientific theories, art, philosophy and of course history. I believe the BBC has created feeds for the podcast version under subjects. Check the one on baltic crusades as its a good intro into the subject and funny too.

You'll be delighted to hear that we have our own podcast which features interviews with subject experts from every corner of history, which can be found here!

Well my two faves fall outside of the time period you requested, but History of Rome and Revolutions by Mike Duncan are both fantastic. The first is pretty self explanatory in what it covers, going from the founding myth of Rome to the fall of the Western Empire in the 400's. Revolutions covers nearly a dozen major revolutions , starting with the English revolution in the 1640's to the Russian Revolution of 1917.

I came here looking for recommendations, but this post is recent enough that I figured I'd leave a couple of history podcasts I enjoy that briefly touch on the medieval period. Neither of them primarily focus on it, and instead have a broader focus on cultural history. Both are produced by people with academic backgrounds in history.

Sects Ed is a very well researched podcast on the history of peripheral religious groups. They have two episodes that specifically focus on medieval and early Renaissance European heterodoxies (episode 2, on the Cathars, and episode 13, on Girolamo Savonarola). Sadly, they haven't done an episode since 2018, but their backlog is great.

Bad Gays focuses on historical figures who engaged in same-sex and/or gender non-conforming activity, but lived morally troubling or complex lives and have therefore been overlooked in popular accounts of queer history. Most episodes focus on the 19th and 20th centuries, but they do have a medieval episode (season 2, episode 9, on Piers Gaveston) as well as a Renaissance episode (season 2, episode 3, on Pietro Aretino). The hosts hold a social constructivist view of sexuality, arguing that sexuality is culturally contingent, so the categories we use today are not universal and cannot be projected onto the past. Some of the online reviews mistake this as erasing queerness from history, but they are very clear that this extends to heterosexuality (as argued by gay historian Jonathan Ned Katz in The Invention of Heterosexuality).

Edited a capitalization error.