This is a survey paper on the moduli spaces of sheaves and related enumerative problems. We begin with an overview of the moduli theory such as a stack of coherent sheaves, Gieseker semistability, and moduli spaces. The rest of the paper focuses on geometries and enumerative problems coming from the moduli spaces of sheaves on curves and surfaces. In particular, we introduce the topology of the moduli spaces, Verlinde formulas, Hilbert scheme of points and curves, Vafa-Witten invariants, and Gopakumar-Vafa invariants.