While NASA’s Kepler mission has identified thousands of planets and measured their periods and radii through transit observations. Planet mass and eccentricity measurements, in addition to periods and radii, are required to constrain planets’ compositions and dynamical histories. Unfortunately, for many Kepler planets such measurements are not feasible or require intensive follow-up observations. In multi-planet systems, however, mutual gravitational perturbations cause variations in the planets’ times of transit that can be used to recover dynamical information about the planetary system. However, inferring planet masses and orbits from these transit timing variations (TTVs) is a difficult inverse problem with a large parameter space and multiple degeneracies. I will describe our approach to TTV inversion that combines Markov chain Monte Carlo simulations with insights from simplified analytic formulae. I will present mass and eccentricity measurements from a study of 145 planets and discuss the implications of these measurements for planets’ compositions and dynamics.