We present detailed yet largely analytical models for gain, optical bandwidth, and saturation power of vertical-cavity laser amplifiers (VCLAs) in reflection and transmission mode. VCLAs are potential low-cost alternatives to in-plane laser amplifiers and they have the inherent advantage of polarization insensitivity, high fiber coupling efficiency, and low noise figure. Simple formulas for the optical gain-bandwidth product are derived which are valid for any type of Fabry-Perot amplifier. Our saturation model is based on rate equations for electrons and holes. It considers a sub-linear material gain, gain enhancement by the standing wave effect, Auger recombination, defect recombination, and spontaneous emission. Common linear approximations are avoided to correctly predict performance limits. Excellent agreement with measurements on novel 1.3-micron VCLAs is obtained. The models are used to analyze device performance and to investigate optimization options. With reduced top mirror reflectivity and increased pump efficiency, substantial and simultaneous improvements of optical bandwidth and saturation power are predicted without sacrificing gain. Parameter plots are given which allow for an easy exploration of the VCLA design space, matching desired performance goals with the required mirror reflectivity and pump current.