In this thesis, higher derivative theories are investigated. Ostrogradskian instability of higher derivative theories is examined both at the classical and quantum levels. It is shown that avoiding the instability in nondegenerate higher derivative theories is impossible. Moreover, the degenerate model of relativistic particle with a curvature term is studied as a viable higher derivative theory. Most of the work we present here is not original. We give a review of the literature and compile various detached works that already exist.