Now that every browser has a built-in password manager, you don't even have to remember the passwords.
You're daft if you're going to rely on your browser to manage passwords. Much easier to lose them that way then forgetting them - use a separate password manager at minimum, one that isn't tied to a particular product or - worse - a particular instance of a product.
As tooki notes above, Safari (which I use) uses the Mac Keychain, so log-in/passwords are available not only to the browser but to any application which takes advantage of the Keychain. And the Keychain synchronizes across all of your devices, linked by an Apple ID. And it works. If I create or change a password for, say, my bank's website, the Keychain on my laptop (what I'm using now) is updated and then that change is pushed to other devices. When I log in using my desktop the new password is there.
That said, my only issue with the Keychain is that it can't be shared. This means that if my wife changes the password for the AMZN account it updates only her keychain, not mine, so she has to tell me the new password. To get around that, we also use 1Password. Having two separate, parallel password managers means that if for some reason one of the databases is lost, the other one still exists. (This is the other "gotcha" mentioned by tooki.)
I understand why Firefox and Opera all have their own password managers and synchronization schemes, but it would be nice if they let you take advantage of operating-system features.