Of all the fundamental fermion masses, those of the neutrinos alone remain unmeasured. From their unknown origin to their effects on the evolution of the universe, neutrino masses are of interest across cosmology, nuclear physics, and particle physics. Neutrino oscillation experiments have set a non-zero lower limit on the mass scale, in contradiction to the original Standard Model prediction. To measure neutrino mass precisely one must turn to beta decay and search for a telltale distortion in the spectrum. I will describe a new technique called Cyclotron Radiation Emission Spectroscopy (CRES), in which beta decay of tritium occurs in a magnetic field and each electron's ~1 fW of cyclotron radiation is directly detected. Electron energies are then determined via a relativistic relationship between energy and frequency. I will present the first CRES-based mass limits from the Project 8 experiment, which demonstrate the promise of this technique for surmounting the systematic and statistical barriers that currently limit the precision of direct neutrino mass measurements. I will also describe the next steps on the path to sensitivity to a mass of 40 meV/c^2, covering the entire inverted ordering of neutrino masses.