iOS: There are a lot of fantastic podcast apps for the iPhone (Downcast is our favourite). If you're looking for one that does things a little differently, Castro is a new podcast manager that uses gestures for navigation, has a great iOS 7 look, and includes a lot of options for data management.
Castro has all the features you'd expect from a podcast manager. You can easily subscribe to podcasts, manage the episodes you have, and stream any episode you don't have. You can also set a variety of playback speeds, create download rules (like Wi-Fi only), and easily customise the settings for each podcast you listen to.
What's really nice about Castro is that you can customise the storage space that the app is allowed to have. In the app's settings, you can set download limit sizes and cache sizes. From there, Castro will intelligently delete old episodes and restrict downloads. If you find yourself short on storage space on your phone, this is a fantastic feature. Castro is missing some features like playlist and sleep timers, but if those advanced features don't matter to you, Castro's a fantastic podcast app that's worth a look.
Castro ($2.99) [iTunes App Store]