The revelation principle is a key tool in mechanism design. It allows the designer to restrict attention to truthful mechanisms, greatly facilitating analysis. This is also borne out algorithmically, allowing certain computational problems in mechanism design to be solved in polynomial time. Unfortunately, when not every type can misreport every other type (the partial verification model) or—more generally—misreporting can be costly, the revelation principle can fail to hold. This also leads to NP-hardness results. The primary contribution of this article consists of characterizations of conditions under which the revelation principle still holds when reporting can be costly. (These are generalizations of conditions given earlier for the partial verification case [11, 21].) Furthermore, our results extend to cases where, instead of reporting types directly, agents send signals that do not directly correspond to types. In this case, we obtain conditions for when the mechanism designer can restrict attention to a given (but arbitrary) mapping from types to signals without loss of generality.