Simplicial volume is a homotopy invariant of oriented closed connected manifolds introduced by Gromov in his proof of Mostow rigidity. On the one hand, the simplicial volume of a Riemannian manifold encodes non-trivial information about the Riemannian volume; on the other hand, simplicial volume can be described in terms of a certain functional analytic version of homological algebra (bounded cohomology). In this article we survey important properties and applications of simplicial volume as well as useful techniques for working with simplicial volume. 53C23, 57R99 1. Definition and history Simplicial volume is a homotopy invariant of oriented closed connected manifolds that was introduced by Gromov in his proof of Mostow rigidity [31][13]. Intuitively, simplicial volume measures how difficult it is to describe the manifold in question in terms of simplices (with real coefficients): Definition 1.1. (Simplicial volume) Let M be an oriented closed connected manifold of dimension n. Then the simplicial volume of M (also called the Gromov norm of M) is defined as M := [M ] 1 = inf |c| 1 c ∈ C n (M ; R) is a fundamental cycle of M ∈ R ≥0 , where [M ] ∈ H n (M ; R) is the fundamental class of M with real coefficients. • Here, | • | 1 denotes the 1-norm on the singular chain complex C * (• ; R) with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space X and a chain c = k j=0 a j • σ j ∈ C * (X; R) (in reduced form), the 1-norm of c is given by |c| 1 := k j=0 |a j |. • Moreover, • 1 denotes the 1-semi-norm on singular homology H * (• ; R) with real coefficients, which is induced by | • | 1. More explicitly, if X is a topological space and α ∈ H * (X; R), then α 1 := inf |c| 1 c ∈ C * (X; R) is a cycle representing α. Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.