The research area of this thesis is operator-algebraic or non-commutative probability theory and more specifically non-commutative Markov processes, which describe the time evolution of a large class of open quantum systems. One usual method in non-commutative probability is to take established tools from classical probability and see how they can be generalized to the operator-algebraic methods. In this work, we apply this philosophy to the theory of topological Markov processes, which are a concept from symbolic dynamics and coding theory. In this context, the word "topological" means that we only describe which trajectories, or sequences of system states, are possible for given dynamics, without tracking the probabilities of a certain trajectory. We lift this idea to the theory of non-commutative Markov processes, by describing quantum system by their topological properties. This gives us a non-deterministic but non-probabilistic description of the dynamics. We demonstrate that many commonly considered stochastic properties of such dynamics, especially those relevant to the asymptotic behaviour of the system, can be derived completely from this topological description. The central new concept of this thesis are reach maps, maps on orthogonal projections of an algebra that capture the topological essence of completely positive operators. They turn out to be a useful concrete representation of the previously vague concept of a "topological Markov operator" which we were looking for. Reach maps encode which sequences of events are possible in given dynamics and are exactly the right morphisms to form a category in which we can express non-deterministic topological dynamics. To define reach maps, we apply methods from non-commutative topology, which uses the universal enveloping von Neumann algebra to apply von Neumann algebra methods to C*-algebras, bringing measure theoretic and topological objects closer to each other. We adapt this theory to our cause by generalizing some of its foundations to admit other enveloping von Neumann algebras than the universal one. In addition to giving definitions for non-commutative topological dynamics in the form of reach maps and a topological Markov condition, main results in this thesis are a characterization of reach maps via cross-ratios from projective geometry, applying Perron-Frobenius theory to reach maps and the discovery of the surprisingly elegant structure of reach maps of conditional expectations.