This paper describes a non-contact methodology for computing river discharge based on data collected from small Unmanned Aerial Systems (sUAS). The approach is complete in that both surface velocity and channel geometry are measured directly under field conditions. The technique does not require introducing artificial tracer particles for computing surface velocity, nor does it rely upon the presence of naturally occurring floating material. Moreover, no prior knowledge of river bathymetry is necessary. Due to the weight of the sensors and limited payload capacities of the commercially available sUAS used in the study, two sUAS were required. The first sUAS included mid-wave thermal infrared and visible cameras. For the field evaluation described herein, a thermal image time series was acquired and a particle image velocimetry (PIV) algorithm used to track the motion of structures expressed at the water surface as small differences in temperature. The ability to detect these thermal features was significant because the water surface lacked floating material (e.g., foam, debris) that could have been detected with a visible camera and used to perform conventional Large-Scale Particle Image Velocimetry (LSPIV). The second sUAS was devoted to measuring bathymetry with a novel scanning polarizing lidar. We collected field measurements along two channel transects to assess the accuracy of the remotely sensed velocities, depths, and discharges. Thermal PIV provided velocities that agreed closely (R^2 = 0.82 and 0.64) with in situ velocity measurements from an acoustic Doppler current profiler (ADCP). Depths inferred from the lidar closely matched those surveyed by wading in the shallower of the two cross sections (R^2 = 0.95) but the agreement was not as strong for the transect with greater depths (R^2 = 0.61). Incremental discharges computed with the remotely sensed velocities and depths were greater than corresponding ADCP measurements by 22% at the first cross section and < 1% at the second.