In many applications, the sensitivity of solutions of PDE's to certain parameters in the equations is an important topic. There has been a great deal of work, especially in the chemical engineering field, on sensitivity analysis using direct linear algebra. However, when analyzing PDE's in three space dimensions, the use of direct linear algebra is not always feasible. In this talk, we will discuss a new class of algorithms which use matrix-free methods to solve all linear systems. Numerical examples from CFD and groundwater flow will be presented.