This paper presents the design, analysis, performance evaluation, and preliminary experimental validation of a globally asymptotically stable (GAS) filter for simultaneous localization and mapping (SLAM) with application to unmanned aerial vehicles (UAVs). The SLAM problem is formulated in a sensor-based framework and modified in such a way that the system structure may be regarded as linear time-varying for observability purposes, from which a Kalman filter with GAS error dynamics follows naturally. The proposed solution includes the estimation of both body-fixed linear velocity and rate-gyro measurement biases. Both simulation results and preliminary experimental results, using an instrumented quadrotor equipped with a RGB-D camera, are included in the paper to illustrate the performance of the algorithm under realistic conditions.