In recent years, methods from algebraic topology in pure mathematics have been applied to model parallelism in computer networks and in concurrent software, and its effects on the complexity of coordination algorithms. A non-concurrent execution of a program is classically modelled as a (1-dimensional) directed graph. A concurrent execution should however be seen as a trajectory on a multi-dimensional space - one dimension per process. Moreover, that space has to model restrictions due to synchronizations and the direction of the time-flow. Particularly attractive models are the so-called higher-dimensional automata, which can be described as cubical complexes with a (local) partial order or as directed spaces. It is the aim of the project to combine mathematical methods from homotopy theory and geometry with combinatorial methods in the study of these automata, their properties and their limitations with respect to parallel computations. Current research attempts to apply modified homotopy notions to the classification of both executions and states. This includes a covering theory for model spaces and a study of the ``components'' of the fundamental category of a directed space. Notions and methods from category theory and relativity theory adjusted to the situation have shown to be useful. Though the methods are quite general and theoretically minded, possible applications range from the understanding of distributed databases to program analysis and verification since they provide new tools in attacking the state space explosion problem. Cooperation with Éric Goubault and Emmanuel Haucourt, CEA,Saclay, France, and Stefan Sokolowski, Academy of Sciences, Gdansk, Poland