The curvature of smooth curves in \(R^n\) is well-defined and can be directly computed using smooth arc length parametrizations. The goal of this paper is to understand and develop definitions of curvature on arbitrary Riemannian manifolds. We begin with a brief understanding of curvature on surfaces in \(R^3\), notably the Gaussian curvature and the mean curvature. In abstract Riemannian manifolds, we use the associated metric to construct a directional derivative that properly defines the acceleration of a curve within the manifold. Geodesics, or curves whose acceleration is identically zero, are defined as being flat, similar to straight lines in \(R^n\). The Riemann curvature tensor \(R\) is a measure of the local flatness of a Riemannian manifold, where a manifold is locally isometric to Euclidean space if and only if its curvature tensor vanishes identically. From \(R\) we obtain the Ricci curvature tensor and the scalar curvature. We conclude by defining the sectional curvature, which allows us to analyze the curvature of hypersurfaces using the extension of the Gaussian curvature to arbitrary 2-submanifolds.