The dimension of the attractor of a dynamical system is a quantity which is closely related to properties as the systems stability or chaotic behaviour. Various definitions of dimensions have been established which may also take noninteger values. The most important ones are Hausdorff and box dimension. We investigate the relationships between various types of dimension and the geometric complexity of invarariant sets (e.g. isolated equilibria, cycles or Cantor-type sets). We consider both smooth flows and discrete-time dynamical systems on Riemannian manifolds. The results are formulated in terms of the singular values of the linearized evolution operator and curvature properties of the manifold. The hyperbolic or quasi-hyperbolic structure of invariant sets is used in dimension estimates and entropy terms are introduced in dimension bounds. Additionally global informations are considered such as the homology groups of the manifold, natural Lyapunov functions and Losinskii norms. On the basis of Hausdorff dimension estimates generalizations of certain global stability results of Hartman-Olech and other types of classical theorems from the Bendixson-Poincaré theory are derived. The box dimension gives important hints at the possibility to use embedding strategies. In concrete physical or technical systems the considered maps are often not only non-smooth but besides this even non-injective. For a class of uniformly non-injective maps it is possible to include into the dimension estimates the degree of non-injectivity. For the first time such estimates using non-injectivity information have been considered by A. Franz. We apply this approach to a class of piecewise smooth maps.