One of the primary functions I perform on my smartphones is listening to podcasts, which I actually listen to much more than music. Nokia has an excellent Nokia Podcasting application that lets you subscribe and download podcasts with and the iPhone has iTunes integration so syncing podcasts over is fairly simple. There are a couple of 3rd party apps for Android, but I was very pleased to see Google launch their own podcatcher called Listen. Listen lets you search, subscribe, download, and stream podcasts to your Android-based device.
One awesome aspect of having a native Google podcatcher is that the Google search bar/box is used, along with Google servers, to find podcasts so you should be able to find more than you could through a loaded podcasting searching database. After you find a podcast you can press on the result to start streaming or press and hold to see more options, such as downloading for later listening.
It is still early in the beta cycle so they don't yet support scheduled downloads, non-English audio podcasts, and a few more features that are under development.