Lagrangian coherent structures (LCSs) are influential material surfaces that create clearly recognizable patterns in passive tracer distributions advected by an unsteady flow. LCSs can be classified as hyperbolic (locally maximally attracting or repelling material surfaces), elliptic (material vortex boundaries), and parabolic (material jet cores). These surfaces are generalizations of classical invariant manifolds, known in dynamical systems theory, to finite-time unsteady flow data. This Lagrangian perspective on coherence is concerned with structures formed by fluid elements, as opposed to the Eulerian notion of coherence, which considers features in the instantaneous velocity field of the fluid. Various mathematical techniques have been developed to identify LCSs in two- and three-dimenisonal data sets, and have been applied to laboratory experiments, numerical simulations and geophysical observations.