The dynamical interactions of our Solar System have been studied in depth,since Isaac Newton recognized that the planets may not be stable to each other’s,gravitational perturbations. Recently, the discovery of exoplanet systems, including,approximately a thousand planet candidates in systems of more than two bodies, has,opened an extremely vast and diverse laboratory for planetary dynamics. In this,dissertation, I describe techniques for measuring the dynamical, post-Keplerian,interactions of planetary systems. Such signals often require numerical N-body,analysis and photodynamic techniques combined with Bayesian statistics to,correctly determine the properties of the planetary systems causing them. By,simultaneously fitting the entire lightcurve data set at once, I am able to extract low,signal-to-noise effects such as the resonance dynamics of a very faint system,(Kepler-223), the slow orbital precession of a giant planet system (Kepler-108), and,transit timing variations among very small and low mass planets (Kepler-444). I use,these analyses to gain physical insight into the system’s history, such as Kepler-108's ,potentially chaotic, violent past. Kepler-223's present structure indicates a,migration origin for at least some close-in, sub-Neptune planets, which I explore in,terms of tidal dissipation, smooth and stochastic migration, and secular evolution. I,also analyze circumbinary systems including the newly discovered KIC 10753734.,Taken together, these results provide insight into planetary formation in a broad array ,of environments for planet from compact sub-Neptune systems to Jupiters and ,circumbinary planets.