We will discuss recent results on optimal stopping problems in which the state processes are governed by stochastic functional differential equations as well as stochastic differential equations with random coefficients. In particular, the existence and uniqueness for the viscosity solution of the Hamilton‐Jacobi Bellman variational inequality (HJBVI) in a Banach space and the adapted solution for the backward stochastic partial differential variational inequality (BSPDVI) will be examined. In addition, some challenging open problems will be mentioned in this talk.