Abstract: Complex algebraic plane curves are the zero sets of algebraic polynomials in the complex (projective) plane. The topology and realizability of singular curves has been studied classically but open problems remain to this day. We will start out exploring the topology of complex plane curves through examples with explicit polynomials. Symplectic curves generalize complex curves - they share some important properties but also have much greater flexibility. We will introduce these objects and some guiding research questions about symplectic curves. Then, we will compare properties of complex and symplectic plane curves, discuss open problems, and survey historical and recent results.