This paper gives definitions and results about password-based protocols for authenticated key exchange (AKE), mutual authentication MA), and the combination of these goals (AKE, MA). Such protocols are designed to work despite interference by an active adversary and despite the use of passwords drawn from a space so small that an adversary might well enumerate, off line, a user's password. While several such password-based protocols have been suggested, the underlying theory has been lagging, and some of the protocols don't actually work. This is an area strongly in need of foundations, but definitions and theorems here can get overwhelmingly complex. To help manage this complexity we begin by defining a model, one rich enough to deal with password guessing, forward secrecy, server compromise, and loss of session keys. The one model can be used to define various goals. We take AKE (with implicit authentication---no one besides your intended partner could possibly get the key, though he may or may not actually get it) as the basic goal. Then we prove that any secure AKE protocol can be embellished (in a simple and generic way) to also provide for MA. This approach turns out to be simpler than trying to augment an MA protocol to also distribute a session key. Next we prove correctness for the idea at the center of the Encrypted Key-Exchange (EKE) protocol of Bellovin and Merritt: we prove (in an ideal-cipher model) that the two-flow protocol at the core of EKE is a secure AKE. Combining with the result above we have a simple 3-flow protocol for AKE,MA which is proven secure against dictionary attack.