The electromagnetic (EM) inverse problem in geophysics consists of deducing the Earth’s conductivity from measurements of electric or magnetic fields near the surface. It requires fast and accurate forward modeling, a method of solving nonlinear equations, and a method of controlling nonuniqueness. Four modeling methods are considered for simulation of airborne EM data over a 3-D conductivity model: a one-dimensional (1-D) approximation, the 3-D Born approximation, a finite-element (FE) method with approximate boundary conditions, and an FE method with exact boundary conditions. The 1-D and Born approximations are fast, but not very accurate. The two FE methods give similar responses, and the method with approximate boundary conditions requires much less computer resources. Three methods are considered for solving the EM inverse problem by unconstrained nonlinear optimization: conjugate gradient (CG), quasi-Newton, and Gauss-Newton (GN) methods. For the simple models considered, the GN method required the least CPU time; the CG method, the most. An alternative to regularization for controlling nonuniqueness is joint inversion of several different surveys to produce a single model that fits all of the data. An example with airborne EM data and pole-pole dc resistivity data shows that such joint inversion can effectively reduce nonuniqueness in EM inversion.