Download and install the official Podcasts app on your iPhone to start listening to podcasts. The Podcasts app lets you both download and stream podcasts from anywhere with a Wi-Fi or cellular data connection. You can also use the iTunes application on your computer to download podcasts and then transfer them by syncing your iPhone.
Downloading Podcasts on Your iPhone
Open the App Store on your iPhone, browse to the Podcasts app (link in Resources) and tap the "Free" button to install it. Once the app is on your phone, open it and browse or search for the podcast you want to download. Tap the podcast to view its Details page and then tap the "Download" icon next to the episode you want to download. You can also select the "Subscribe" button to receive the most recent episodes in your podcast feed. When the download is complete, it appears in the My Podcasts section of the Podcasts app. Tap the title to listen to it.
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If you don't want to save each podcast episode on your iPhone, stream it directly through the Podcasts app. Tap the podcast you want to stream and then tap the name of the episode rather than the Download button. The app immediately streams the episode.
Downloading Using ITunes
You can also download podcasts in the iTunes application on your computer and sync them to your iPhone. Open the iTunes store on your computer, browse to the podcast you want to download and click the "Free" button to download it. Click "Subscribe" to subscribe to the podcast and automatically download all future episodes.