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A slice is an intersection of a hyperplane and a self-similar set. The main purpose of this work is the mathematical description of slices. A suitable tool to describe slices are branching dynamical systems. Such systems are a generalisation of ordinary discrete dynamical systems for multivalued maps. Simple examples are systems arising from Bernoulli convolutions and beta-representations. The connection between orbits of branching dynamical systems and slices is demsonstrated and conditions are derived under which the geometry of a slice can be computed. A number of interesting 2-d and 3-d slices through 3-d and 4-d fractals is discussed.