Wallet support for Monero is limited. It is very likely caused by its different underlying technology, compared to Bitcoin and other popular cryptocurrencies. Ledger hardware wallet has Monero on its roadmap.
MyMonero is the only web wallet for Monero. It was created by Riccardo Spagni, the lead developer of Monero, who also oversees its development. The wallet works like any account, where you login online to access your funds. MyMonero encrypts your keys and stores the data on the servers. Only you have access to your funds.
The official wallet and the official full node client in one. It can be downloaded from the official Monero website.
XMRWallet is an open-source, free Monero wallet that enables you to send and receive Monero without having to download any software. It is open-source, which means anyone can see the code and help make it better. Plus, you do not need to register or give any of your personal information to use it. XMRWallet.com does not keep any records of your transactions. You can use it without worrying about anyone seeing what you are doing with your money.
Cryptocurrency wallets, like the ones below, help you store your Monero coins so they're not exposed on an exchange.