Let X be a complete separable bounded metric space and \(\mu\) a Borel probability measure on the space \(Con(X)^ N\) of all N-tuples of contractions of X with the topology of pointwise convergence. Then there exists a unique \(\mu\)-self-similar probability measure \(P_{\mu}\) on the space \({\mathcal K}(X)\) of all non-empty compact subsets of X. Here a measure P on \({\mathcal K}(X)\) is called \(\mu\)-self-similar if, for every Borel set \(B\subset {\mathcal K}(X)\), \[ P(B)=\int P^ N((K_ 0,...,K_{N-1})| \cup^{N-1}_{i=0}S_ i(K_ i)\in B)d\mu (\quad S_ 0,...,S_{N- 1}). \] If, for \(\mu\)-a.e. \((S_ 0,...,S_{N-1})\), each \(S_ i\) has an inverse which satisfies a Lipschitz condition then there is an \(\alpha\geq 0\) such that, for \(P_{\mu}\)-a.e. \(K\in {\mathcal K}(X)\), the Hausdorff dimension H-dim(K) is equal to \(\alpha\). If \(X\subset {\mathbb{R}}^ d\) is compact and has non-empty interior and if \(\mu\)-a.e. \((S_ 0,...,S_{N- 1})\) consists of similarities which satisfy a certain disjointness condition w.r.t. X then \(\alpha\) is determined by the equation \[ \int \sum^{N-1}_{i=0}Lip(S_ i)^{\alpha}d\mu (S_ 0,...,S_{N-1})=1, \] where \(Lip(S_ i)\) denotes the (smallest) Lipschitz constant for \(S_ i\). Under fairly general assumptions the \(\alpha\)-dimensional Hausdorff measure of \(P_{\mu}\)-a.e. \(K\in {\mathcal K}(X)\) equals 0. If \(\mu\) and X are chosen in a rather special way then \(P_{\mu}\)-a.e. \(K\in {\mathcal K}(X)\) is the graph of a homeomorphism of [0,1] (or a curve or the graph of a continuous function).