Ok, this will take a while to write out. Here we go

• Dan Carlin's Hardcore History. He's not a professional historian, but he's very engaging on his narrative style and reads a lot before doing an episode. But still, always take it with a pinch of salt, something may slip by him because he's not a historian (he's aware of it and recognises his limits).
• Fall of Civilizations. Pretty new one, it's good for getting an overall picture of a long history.
• History of Philosophy Without any Gaps. Counts more like a philosophy podcast, but here we go. It's done by Peter Adamson, a philosopher from King's College, London. And it's really cool that after a few episodes on each philosopher, he interviews an expert on that philosopher, so it is always extremely well informed.
• History on Fire. Daniele Bolleli is a historian, but I am not 100% sure about how legit he is. I take this one always with a cautionary pinch of salt, just like Hardcore History. But it is very engaging.
• In Our Time: History. Great BBC podcast, basically 30-min interviews with a couple of experts on a random history theme.
• NT Pod. It's a podcast about a historical critical look at the New Testament. It's done by Mark Goodacre, from Duke University, one of the few scholars that believes that the synoptic problem should be solved without the hypotetical Q.
• Pax Britannica. Samuel Hume is a historian who knows what he's talking about, and he does a pretty good job of telling the history of the British Empire.
• History of Rome and Revolutions by Mike Duncan are 2 of the best history podcasts out there. He started with History of Rome, and when finished went on with Revolutions. He's well informed, and his style is very engaging. Also, he's studied politics, so his explanations tend to focus on that side of things.
• Russian Rulers. It's done by a historian, it's well informed, and despite the sound quality in the first episodes it's worth listening to.
• History of English. It's a cool one, talking about the origins and evolution of the language, and of the people that spoke it. I am learning more than I expected from it.

There are more than I listen to, but these are the ones that I would recommend completely without hesitation. If it is your thing, then, well... enjoy. :)

If you are interested in history podcasts, I listen to quite a few of them myself, I will come back with a list of good ones.

This is awesome, thank you very much for this. I would probably add Coursera to the Online Education section, and there are also lots of History podcasts of good quality. A good place to start is the AskHistorians podcast, created by the good people at the AskHistorians subreddit, where they interview actual historians about their field of work.