In this talk, I will discuss curves with self-intersections on hyperbolic surfaces. In the first part, I will speak about the k-systoles (where k is a natural number) of hyperbolic surfaces, which are the shortest closed geodesics with at least k self-intersections. In the second part, I will discuss how results regarding self-intersections of closed curves can be extended to arcs. This talk is based on joint work with A. Basmajian, M. Doan, H. Parlier, and B. Xu.