The aim of this project is to explore the geometry and topology of moduli spaces of Riemann surfaces and monopoles. These are fundamental to modern mathematical physics and used as a tool to study path integrals in quantum field theory. We aim to make progress in the field to produce concrete results, which will provide further insight into the general structure of moduli spaces. We will develop new techniques to study the moduli spaces of Riemann surfaces arising from a new and interesting way of counting lattice points in the moduli space and from cone moduli spaces. We will also develop a rigorous framework for studying the moduli space of monopoles arising in geometric Langlands.