The work at hand discusses various universality results for locally univalent and conformal metrics. In Chapter 2 several interesting approximation results are discussed. Runge-type Theorems for holomorphic and meromorphic locally univalent functions are shown. A well-known local approximation theorem for harmonic functions due to Keldysh is generalized to solutions of the curvature equation. In Chapter 3 and 4 these approximation theorems are used to establish universality results for locally univalent functions and conformal metrics. In particular locally univalent analogues for well-known universality results due Birkhoff, Seidel & Walsh and Heins are shown. / In Kapitel 2 werden Runge-Sätze für holomorphe und meromorphe lokal schlichte Funktionen und ein lokaler Approximationsstaz für konforme Metriken mit negativer Krümmung bewiesen. Mithilfe dieser Sätze werden In Kapitel 3 und 4 Universalitätsresultate für lokal schlichte Funktionen und konforme Metriken gezeigt.