There are many applications for which small unmanned aerial vehicles (SUAVs) are well suited, including surveillance, reconnaissance, search and rescue, convoy support, and short-range low-altitude perimeter patrol missions. As technologies for microcontrollers and small sensors have improved, so have the capabilities of SUAVs. These improvements in SUAV performance increase the possibility for hazardous missions through mountainous and urban terrain in the successful completion of many of these missions. The focus of this research was on remote terrain navigation and the issues faced when dealing with limited onboard processing and limited payload and power capabilities. Additional challenges associated with canyon and urban navigation missions included reactive path following, sensor noise, and flight test design and execution. The main challenge was for an SUAV to successfully navigate through a mountainous canyon by reactively altering its own preplanned path to avoid canyon walls and other stationary obstacles. A robust path following method for SUAVs that uses a vector field approach to track functionally curved paths is presented along with flight test results. In these results, the average tracking error for an SUAV following a variety of curved paths is 3.4~m for amplitudes ranging between 10 and 100~m and spatial periods between 125 and 500~m. Additionally, a reactive path following method is presented that allows a UAV to continually offset or bias its planned path as distance information from the left and right ranging sensors is computed. This allows the UAV to to center itself between potential hazards even with imperfect waypoint path planning. Flight results of an SUAV reactively navigating through mountainous canyons experimentally verify the feasibility of this approach. In a flight test through Goshen Canyon in central Utah, an SUAV biased its planned path by 3 to 10~m to the right as it flew to center itself through the canyon and avoid the possibility of crashing into a canyon wall.