Numberphile podcast by Brady Haran
My Favorite Theorem by Kevin Knudson and Evelyn Lamb (in each interview, a mathematician shares their favorite theorem)
The Joy of $x$ by Quanta Magazine. Steven Strogatz interviews scientists (not exclusively mathematicians, but so far, including a disproportionately high number of mathematicians)
Mathematical Moments from the AMS. Based on the posters (flyers?) that the AMS has been distributing for many years.
Mathematically Uncensored "a podcast where our talk is real and complex but never discrete. Hosted by Dr. Pamela E. Harris and Dr. Aris Winger. Join Pamela and Aris for a conversation on the current events and issues for minorities in the mathematical sciences."