A high number of transneptunian objects are part of binary systems. By studying the orbits of these binaries, a great deal can be revealed about the objects’ formation, mass, and other physical properties. However, with ever increasing observational baselines and improvement of modeling techniques, a new generation of orbit analyses is now possible. This new generation of analyses, using advanced statistical techniques and cutting-edge orbital integrators, can place valuable constraints on the shapes and spin poles of transneptunian binaries. Core to this new generation of analyses is the inclusion of non-Keplerian effects caused by the aspherical shapes of transneptunian binaries. To do this, we have developed MultiMoon, a non-Keplerian astrometric orbit fitting tool. Using MultiMoon, we have begun to explore the shapes and spin poles of a few transneptunian binaries, including Haumea, Manwe-Thorondor, 2002 UX25, Salacia-Actaea and others. In this presentation, we will share some of the key results of our ongoing analyses and discuss some unique insights into these complex dynamical systems.