Introduction. The topological structure of certain subsets of operator algebras has been studied since the beginning of spectral theory. However, the viewpoint of differential geometry has only recently been adopted. Several papers, most of them included in the references, have appeared in the last years which describe geometrical invariants of certain sets of operators: projections, selfadjoint invertible operators, positive invertible operators, relatively regular operators, and so on. More generally, one can study the geometry of sets of spectral measures, C∗-algebra and group representations, completely bounded maps between C∗-algebras, and so on. It turns out that some deep results, e.g. Haagerup’s theorem on the similarity orbit of cyclic representations [32], find a natural setting in this geometrical viewpoint (see [3]). This paper surveys the results discussed in those works. We omit most proofs but we present some simplifications of the original presentations. There is a rather complete panorama of the work of E. Andruchow, A. Maestripieri, H. Porta, L. Recht, D. Stojanoff and the author, but we also discuss contributions by C. J. Atkin, B. Gramsch, K. Lorentz, M. Martin, S. Semmes and D. R. Wilkins. Warning: even when we study differential geometry of certain subsets of noncommutative C∗-algebras, this paper contains no results on Connes’ non-commutative differential geometry.