Sturmian sequences (or equivalently cutting sequences, Beatty sequences or regular sequences) are the best discrete approximations to straight lines in . They can be presented by substitutions, but also by continued fractions. For more information see e.g. Chapter 2 of . Multi-dimensional Sturmian words are the best discrete approximations to hyperplanes in for any . The case was studied by Berthé and Vuillon . In a joint paper with Berthé  and in a single author paper (to appear) the relation with substitutions and multi-dimensional continued fractions is worked out. A special case leads to the well known Rauzy fractal. The lecture will give a survey of the developed ideas and obtained results.
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