This American Life: This weekly radioshow (and now podcast) has been broadcasting for over 25 years. Each episode explores a central theme, with usually 3 or 4 different stories.
Freakonomics: This weekly podcast explores the economics behind daily life.
99% Invisible: This podcast explores the importance of design in everyday life.
Science Friday: Weekly radio show and podcast about science news.
Note: When I first started this page several years ago I listed some non-fiction podcasts that I used in my teaching. Some of the criteria I used to select podcasts for use in courses: They need to be high quality in production and information, have reliable non-biased and non-sensationalist information with sources, produce regular content, and have a back catalog available. Since then the podcast field has continued to expand and I cannot list all the podcasts that fulfill these criteria. If you are looking for a podcast to help improve your English, have a look at Google Podcasts or Apple Podcasts.