A variety of inverse problems in PDEs, including identification of coefficients, reconstruction of initial data, estimation of source functions, and discovery of boundary conditions, demand the solution of operator equations or the stable evaluation of operators. Instability is a common feature of these operator equations and therefore special means, known as regularization methods, are required for their numerical solution. We will survey some of the basic theory of Tikhonov's regularization method for inverse problems and present some recent results on the nonstationary iterated Tikhonov method.