  In this survey, we discuss some recent results on free boundary minimal
surfaces in the Euclidean unit-ball. The subject has been a very active field
of research in the past few years due to the seminal work of Fraser and Schoen
on the extremal Steklov eigenvalue problem. We review several different
techniques of constructing examples of embedded free boundary minimal surfaces
in the unit ball. Next, we discuss some uniqueness results for free boundary
minimal disks and the conjecture about the uniqueness of critical catenoid. We
also discuss several Morse index estimates for free boundary minimal surfaces.
Moreover, we describe estimates for the first Steklov eigenvalue on such free
boundary minimal surfaces and various smooth compactness results. Finally, we
mention some sharp area bounds for free boundary minimal submanifolds and
related questions.
