Is there any music app for the iPhone that lets you download music and then listen to it offline?

Is there any music app for the iPhone that lets you download music and then listen to it offline?

Since we have enormous amounts of CDs, I use a combination of iTunes and Amazon Music to seek my HD, then download to my phone - and then delete them ( and if I ever want them back, Amazon just seeks them again off the HD - takes seconds.) I also carry a separate mp3players - has maybe 500 podcasts on it, plus as ton of music - has additional storage cards I can add, - this is all on one card plus the native storage.