The classical Rokhlin lemma is a fundamental result, which lies at the basis of many areas in ergodic theory. Simply put, it says that an aperiodic measure-preserving transformation on a Borel probability space can be approximated by cyclic shifts in a suitable way. The error of such an approximation is expressed as the measure of some exceptional set. Motivated by basic ideas of covering dimension, we define the topological Rokhlin dimension of a homeomorphism on a compact metric space. In this case, the dimensional value expresses how well the given dynamical system can be approximated by shifts on large finite intervals. We will also discuss this concept for $\mathbb{Z}^m$-actions. The main result of the talk is that free $\mathbb{Z}^m$-actions on finite dimensional compact metric spaces have finite topological Rokhlin dimension. This builds on generalizations of the marker property, as introduced by Gutman, and of a technical result by Lindenstrauss. If time permits, I will hint at some C*-algebraic consequences.