I've seen some websites (Medium, Basecamp 3, etc.) that require only an e-mail to sign up or log in to their app. Basically, you enter an e-mail, the app sends a one-off link to your e-mail, you click that link, and you can sign in.
There are a couple of significant benefits to this approach (that I can think of):
a) Often, when I forget password — especially on mobile — I have to enter my e-mail address, check e-mail, go to the click, type the new password twice, and then log in again. This approach removes the password entirely, so you cannot ever forget your password.
b) You cannot compromise customers' passwords and thus accounts since you don't have them.
What are some disadvantages to this approach?