Q: I lost various purchased albums after updating to iOS 8.4.1 and can't download songs from iTunes Store.

Hello there, FedSolo.

First, it sounds like you've been unable to download purchased content to your iPhone. The following Knowledge Base article goes through some great steps for getting to your purchased content and downloading it to your device:

Download past purchases from iTunes - Apple Support

Before you redownload

1. Sign in with the Apple ID that you used for the original purchase. Use these steps to see which Apple ID you used to download an item.
2. Check to see if the purchases you're looking for are hidden. If your purchases are hidden, you might not see them even after you follow the steps below. Learn how to unhide purchases.
3. See if the content is available for redownload. Previously purchased items might not be available if they're no longer on the iTunes Store. Also, make sure that you’re in a country where you can redownload items. If you changed your Apple ID from one country to another, you can’t download items you purchased in a previous country.

If you're a member of Apple Music, items that you've saved for offline use won't appear in your purchased section. Learn more about Apple Music.

If you're using Family Sharing, you can download content purchased by other members of your family group.

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Redownload a song

On an iPhone or iPod touch

1. Open the iTunes Store app.
2. From the menu the bottom of the screen, tap More.
3. Tap Purchased, then tap Music.
4. Tap Not on This iPhone or Not on This iPod. 
5. Scroll to find the song that you want to download, then tap it. 
6. Tap the Download icon  to the right of the title. Your song downloads to your device.

In regards to your second issue, if you haven't found your purchased albums listed in the above steps, follow the instructions found in this article:

See your purchase history in the iTunes Store - Apple Support

If those steps still don't show your purchases:

If you don't recognize a purchase, first check to see if it's one of these types of charges before contacting Apple. 

If you don't see your purchase, it might be hidden. Learn how to hide or unhide items in your Purchase History. 

If you can't find an item that you purchased in the iTunes Store within the last 90 days, report a problem. For older purchases, contact iTunes Store support, and select Purchases, Billing, and Redemption.

Thanks for participating in Apple Support Communities.

Kind Regards.