Choreographic programming is an emerging programming paradigm for concurrent and distributed systems, whereby developers write the communications that should be enacted and then a distributed implementation is automatically obtained by means of a compiler. Theories of choreographic programming typically come with strong theoretical guarantees about the compilation process, most notably: the generated implementations operationally correspond to their source choreographies and are deadlock-free. Currently, the most advanced incarnation of the paradigm is Choral, an object-oriented choreographic programming language that targets Java. Choral deviated significantly from known theories of choreographies, and introduced the possibility of expressing higher-order choreographies (choreographies parameterised over choreographies) that are fully distributed. As a consequence, it is unclear if the usual guarantees of choreographies can still hold in the more general setting of higher-order ones. We introduce Chorλ, the first functional choreographic programming language: it introduces a new formulation of the standard communication primitive found in choreographies as a function, and it is based upon the λ-calculus. Chorλ is the first theory that explains the core ideas of higher-order choreographic programming (as in Choral). Bridging the gap between practice and theory requires developing a new evaluation strategy and typing discipline for λ terms that accounts for the distributed nature of computation in choreographies. We illustrate the expressivity of Chorλ with a series of examples, which include reconstructions of the key examples from the original presentation of Choral. Our theory supports the expected properties of choreographic programming and bridges the gap between the communities of functional and choreographic programming.