This paper presents several ways to compute lower and upperbounds for MaxSAT based on calling a complete SAT solver. Preliminary results indicate that (i) the bounds are of high quality, (ii) the bounds can boost the search of MaxSAT solvers on some benchmarks, and (iii) the upper bounds computed by a Stochastic Local Search procedure (SLS) can be substantially improved when its search is initialized with an assignment provided by a complete SAT solver.