We introduce the formalism of supersymmetric quantum mechanics, including super-symmetry charges,Z2-graded Hilbert spaces, the chirality operator and the Wittenindex. We show that there is a one to one correspondence of fermions and bosons forenergies different than zero, which implies that the Witten index measures the dif-ference of fermions and bosons at the ground state. We argue that the Witten indexis the index of an elliptic operator. Quantization of the supersymmetric non-linearsigma model shows that the Witten index equals the Euler characteristic of the un-derlying Riemannian manifold over which the theory is defined. Finally, the pathintegral representation of the Witten index is applied to derive the Gauss-Bonnettheorem. Apart from this we introduce elementary mathematical background in thesubjects of topological invariance, Riemannian manifolds and index theory / Vi introducucerar formalismen f ̈or supersymmetrisk kvantmekanik, d ̈aribland super-symmetryladdningar,Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin-dexet. Vi visar att det r ̊ader en till en-korrespondens mellan fermioner och bosonervid energiniv ̊aer skillda fr ̊an noll, vilket medf ̈or att Wittenindexet m ̈ater skillnadeni antal fermioner och bosoner vid nolltillst ̊andet. Vi argumenterar f ̈or att Wittenin-dexet ̈ar indexet p ̊a en elliptisk operator. Kvantisering av den supersymmetriskaicke-linj ̈ara sigmamodellen visar att Wittenindexet ̈ar Eulerkarakteristiken f ̈or denunderliggande Riemannska m ̊angfald ̈over vilken teorin ̈ar definierad. Slutligenapplicerar vi v ̈agintegralrepresentationen av Wittenindexet f ̈or att h ̈arleda Gauss-Bonnets sats. Ut ̈over detta introduceras ocks ̊a grundl ̈aggande matematisk bakrundi ämnena topologisk invarians, Riemmanska m ̊angfalder och indexteori.