PSA: Not your keys, not your Monero, even on Kraken
Kraken seems to be the most trusted of CEX's on this sub, but you still shouldn't keep your Monero on the exchange, or this could happen to you:
Essentially, Kraken closes down peoples' accounts if they trip any red flags, but they refuse to tell you why, and a lot of accounts, including mine, have no idea what they did wrong. I created an account, uploaded my driver's license, and bought xmr. Nothing else. But about three days later (with the price of Monero up quite a bit), they forcefully closed my account, and instead of liquidating my holding, they just told me to issue a chargeback. I.E. they stole all of the gains that I made. I should have withdrawn to a wallet as soon as the purchase was made...