Coinbase was one of the pioneers of Bitcoin exchanges. After the Mtgox bankruptcy in 2014 Coinbase became the #1 Bitcoin exchange in the world. Until today Coinbase and their trading platform GDAX (aka Coinbase Pro) are among the leaders in crypto currency exchanges. Coinbase has the probably the best cold storage that any exchange offers, they call it “The Vault”.
Update June 2018: Coinbase has been terminating users accounts without warning, without reasons.