Water megamasers residing in the accretion disks around supermassive black holes (SMBHs) in active galactic nuclei (AGN) provide unique tools for bypassing the distance ladder and making one-step, geometric distance measurements to their host galaxies. The Megamaser Cosmology Project (MCP) is a multiyear campaign to find, monitor, and map such AGN accretion disk megamaser systems, with the goal of constraining the Hubble constant to a precision of several percent. In this talk I will cover the latest results from the MCP. We have systematically applied an updated disk modeling technique to measure the distances to six megamaser-hosting galaxies, which together constrain the Hubble constant to 73.9 +/- 3.0 km/s/Mpc. This value relies solely on maser-based distance and velocity measurements, and it does not use any peculiar velocity corrections. We have explored a variety of different approaches for correcting peculiar velocities, none of which modify this constraint by more than 1-sigma. Our measurement is independent of distance ladders, the cosmic microwave background, and gravitational lenses, and it corroborates prior indications that the local Hubble constant exceeds the early-Universe prediction.