This chapter focuses on the Riemann Mapping Theorem. Building on the Schwarz reflection principle, which is of independent interest, we discuss conformal mappings of domains bounded by rectilinear and circular polygons, and conic sections. Special applications include conformal mappings of doubly connected domains onto annuli, the geometric definition of the elliptic modular function, and an analysis of the boundary behaviour of proper mappings for multiply connected domains. The final section is devoted to the class \(\mathcal S\) of normalized conformal mappings of the unit disc.