This thesis concerns cycles, i.e., topological ovals, in the phase portraits of systems of first order ordinary differential equationsin the plane, with an emphasis on limit cycles, cycles that are isolated from all other cycles. These are of fundamental importance because when asymptotically stable they correspond to limiting periodic behavior in the underlying system of differential equations. We treat basic theorems, with their proofs, concerning existence, non-existence, and unicity of cycles, and culminate with a general theorem guaranteeing existence of an asymptotically stable limit cycle in the phase portrait of systems of first order equations that correspond to differential equations of the form $\ddot x + f(x) \dot x + g(x) = 0$. The thesis includes examples that illustrate the theorems, including the historically important Volterra-Lotka family and the van der Pol oscillator.