Adding to the substantial and significant work done in the last few decades on finiteness conjectures, this monograph focuses on their results in ordinary differential equations and Diophantine geometry, one of the few problems on Hilbert's list to remain unsolved. The authors describe their work in the existence and estimation of finite bounds for the number of limit cycles occurring in certain families of ordinary differential equations, and includes their methods such as bifurcation theory, asymtotic expansions, and methods of differential algebra.