Ocean surface currents and winds are tightly coupled essential climate variables, and, given their short time scales, observing them at the same time and resolution is of great interest. DopplerScatt is an airborne Ka-band scatterometer that has been developed under NASA's Instrument Incubator Program (IIP) to provide a proof of concept of the feasability of measuring these variables using pencil-beam scanning Doppler scatterometry. In the first half of this paper, we present the Doppler scatterometer measurement and processing principles, paying particular attention to deriving a complete measurement error budget. Although Doppler radars have been used for the estimation of surface currents, pencil-beam Doppler Scatterometry offers challenges and opportunities that require separate treatment. The calibration of the Doppler measurement to remove platform and instrument biases has been a traditional challenge for Doppler systems, and we introduce several new techniques to mitigate these errors when conical scanning is used. The use of Ka-band for airborne Doppler scatterometry measurements is also new, and, in the second half of the paper, we examine the phenomenology of the mapping from radar cross section and radial velocity measurements to winds and surface currents. To this end, we present new Ka-band Geophysical Model Functions (GMF's) for winds and surface currents obtained from multiple airborne campaigns. We find that the wind Ka-band GMF exhibits similar dependence to wind speed as that for Ku-band scatterometers, such as QuikSCAT, albeit with much greater upwind-crosswind modulation. The surface current GMF at Ka-band is significantly different from that at C-band, and, above 4.5 m/s has a weak dependence on wind speed, although still dependent on wind direction. We examine the effects of Bragg-wave modulation by long waves through a Modululation Transfer Function (MTF), and show that the observed surface current dependence on winds is consistent with past Ka-band MTF observations. Finally, we provide a preliminary validation of our geophysical retrievals, which will be expanded in subsequent publications. Our results indicate that Ka-band Doppler scatterometry could be a feasible method for wide-swath simultaneous measurements of winds and currents from space.