The representation dimension of an Artin algebra \(A\) has been defined by \textit{M. Auslander} [Representation dimension of Artin algebras. With the assistance of Bernice Auslander. (Queen Mary College Mathematics Notes. London: Queen Mary College) (1971; Zbl 0331.16026)] in several equivalent ways; for example, the representation dimension of \(A\) is the infimum of the global dimensions of algebras \(B\), which are endomorphism rings of \(A\)-modules \(M\) such that \(M\) is both a generator and a cogenerator. Auslander famously characterized algebras of finite representation type by their representation dimension being less than or equal to two. The article under review computes the representation dimension (or upper bounds for it) for various classes of algebras, such as `stably hereditary algebras' (defined in this article as a generalisation of algebras stably equivalent to hereditary ones), certain quotients of self-injective algebras, and incidence algebras of posets. Moreover, it is shown that certain fundamental constructions preserve the representation dimension. In particular, this is true for stable equivalences of Morita type. An attractive corollary is that derived equivalent self-injective algebras must have the same representation dimension. In section six, two conjectures are stated. The first conjecture, which has been sort of folklore, states that the representation dimension of an Artin algebra always is finite. The second conjecture, which came as a surprise, suggests a way of attacking the first conjecture. In fact, it claims a much stronger statement: For each \(A\)-module \(M\) there is another \(A\)-module \(M'\) such that the endomorphism ring \(\text{End}_A(M\oplus M')\) is quasi-hereditary (and hence has finite global dimension). Subsequently, \textit{O. Iyama} [Proc. Am. Math. Soc. 131, No. 4, 1011-1014 (2003; Zbl 1018.16010)] proved the second, and hence also the first, conjecture. \textit{R. Rouquier} [Dimensions of triangulated categories (preprint), available from \url{http://www.math.jussieu.fr/~rouquier/preprints/preprints.html}] then showed that the representation dimension can take arbitrarily large values.