Trace expansions for operator families such as the resolvent, the heat operator and the complex powers are established for elliptic problems containing pseudodifferential elements. We consider operators on closed manifolds, as well as operators on compact manifolds with boundary, where suitable boundary conditions must be added. It is found in general that one can obtain expansions, e.g. of the heat operator trace, in powers t^a and power-logarithmic terms t^a log t, and the stability of the coefficients under perturbations is discussed. A survey is given of the methods relying on pseudodifferential calculus that lead to these results.