The dichotomy spectrum is a crucial spectral notion in the theory of nonautonomous dynamical systems, since it is of significant importance in a stability theory for time-dependent problems, as well as to develop a geometric theory in terms of invariant manifolds or normal forms, or to derive continuation and bifurcation techniques. In this thesis, we mainly study the structure and properties of the dichotomy spectrum for linear difference equations in infinite-dimensional Banach spaces. It is well known that in the finite-dimensional context, the dichotomy spectrum is a union of closed intervals, where the number of intervals is restricted by the dimension of the space and each of these spectral intervals is associated with an invariant vector bundle containing solutions with a certain growth rate. In the general infinite-dimensional situation, one cannot expect such a regular structure. In fact, any compact subset of the positive half-line may occur as a dichotomy spectrum. To overcome this difficulty, we introduce, by means of a measure of noncompactness, two new quantities which we call the upper and lower dichotomy index. These indices are used to decompose the dichotomy spectrum into the lower, essential and upper dichotomy spectrum. Then the main result of this thesis is a Spectral Theorem which describes all possible cases of the upper and lower dichotomy spectrum and yields a "nonautonomous linear algebra" in terms of invariant vector bundles containing solutions with a certain growth rate. More precisely, we show that the lower as well as the upper dichotomy spectrum consists of at most countably many closed spectral intervals. For difference equations satisfying an appropriate compactness property, the dichotomy spectrum is fully determined by the upper dichotomy spectrum. So the Spectral Theorem gives a complete description of the associated dichotomy spectrum in this case. Moreover, we illuminate the relation between the dichotomy spectrum and Lyapunov exponents and deduce suitable characterizations for the above-mentioned dichotomy indices-in the autonomous case the lower and upper dichotomy index coincide with the inner and outer essential spectral radius, respectively. As an application, we transfer our results to continuous time problems and show how the dichotomy spectrum can be used to determine stability properties for nonlinear difference equations.