This thesis is a study of the properties and relationships between isothermal, harmonic and characteristic coordinate systems. The proof of the existence of isothermal and characteristic coordinates on a manifold which is a graph is given using the Uniformization theorem. Equations of prescribed mean curvature are discussed and the relationship between equations of minimal surface type and mean curvature type are shown. It is also proven that the map from a domain parameterized by characteristic coordinates to the domain parameterized by isothermal coordinates is quasiconformal.