With apologies to Marco, AntennaPod is the BEST podcast app on any platform. I especially like that I can point to a local folder and create a pseudo podcast (which, when combined with NewPipe, allows me to pull down lectures from YouTube as audio files to consume as an ad hoc podcast feed).

Sadly... I'm stuck on iPhone and wish someone would port AntennaPod to iOS.

Have you tried fountain.fm?

Yep. Have exchanged emails with Oscar Merry(sp?) about it many times starting from when it was first announced as part of Podcasting 2.0 but the features and layout of AntennaPod are hands-down better. As long as I'm on iOS I'll probably stay with Overcast but I have been using Podcast Addict as well.