It is well known that "hyperbolic" dynamical systems admit a unique invariant probability measure with maximal entropy, which dates back to work of R. Bowen in 1970's. One can naturally ask effectivity of uniqueness for the measure of maximal entropy. This question has received attention over the last decade. In this survey, we review the uniqueness of the measure of maximal entropy for several dynamical systems and recent developments in effective uniqueness for the measure of maximal entropy.