we describe hierarchy of coarse-grained stochastic processes that approximate, on larger scales, microscopic lattice systems simulated by (kinetic) monte carlo algorithm. in applications the microscopic model is often coupled with continuum $pde$ description on larger scales. we present mathematical tools for error estimation of the coarse-graining procedure. a posteriori estimates on the loss of information between the coarse-grained and the microscopic probability measure can guide the coarse-graining procedure at different parts of the computational domain. in specific examples of lattice spin dynamics we demonstrate that coarse-grained $mc$ leads to significant cpu speed up of simulations of metastable phenomena, e.g., estimation of switching times or nucleation of new phases. some recent results obtained in joint work with m. katsoulakis and a. sopasakis will be presented.