We present a study on the power dependence of the microwave surface impedance in thin films of the novel superconductor MgB2. 500 nm thick samples exhibiting critical temperatures ranging between 26 and 38 K are synthesized by an ex-situ post-anneal of e-beam evaporated boron in the presence of an Mg vapor at 900 C. Preliminary results on films grown in situ by a high rate magnetron sputtering technique from stoichiometric MgB2 and Mg targets are also reported. Microwave measurements have been carried out employing a dielectrically loaded niobium superconducting cavity operating at 19.8 GHz and 4 K. The study shows that the electrodynamic response of MgB2 films is presently dominated by extrinsic sources of dissipation, appearing already at low microwave power, likely to be ascribed to the presence of grain boundaries and normal inclusions in the samples.