Enhancement cavities are passive optical resonators in which continuous-wave laser radiation or pulses of a frequency comb are coherently overlapped, allowing for a power and intensity scaling of up to several orders of magnitude. A prominent application is the table-top generation of bright, laser-like radiation in spectral regions where direct laser action is inefficient or not available at all, via intracavity nonlinear optical processes. However, to exploit the full capacity of this technique further progress is needed. This thesis covers central problems of enhancement cavities, such as finding limitations in scaling the circulating power, measuring cavity parameters with high accuracy, tailoring transverse modes and coupling out radiation generated in the cavity. Unprecedented intracavity laser powers were demonstrated, surpassing previous results by an order of magnitude. As an application, harmonics of the fundamental 1040-nm radiation up to the 21st order are generated. Besides reporting these fine experimental results, the thesis provides an excellent introduction into the physics of enhancement cavities, supported by more than 140 references.