Abstract: We study an optimal stopping problem when the state process is governed by a general Feller process. In particular, we examine viscosity properties of the associated value function with no a priori assumption made on the stochastic differential equation satisfied by the state process. Our approach relies on properties of the Feller semigroup. We present conditions on the process under which the value function is the unique viscosity solution to a Hamilton-Jacobi-Bellman (HJB) equation associated with a particular operator. We then apply our results to study viscosity property of optimal stopping problems for some particular Feller processes, e.g. diffusion processes with piecewise coefficients, semi-Markov processes and regime switching Feller diffusions.