It is well known that the analysis of simple waves in an isotropic elastic body simplifies appreciably when the transmitting material is incompressible: the number of simple wave solutions reduces from six to four and, irrespective of the form of the internal-energy function, the governing system of ordinary differential equations can be integrated explicitly. In this paper a general theory of uni-directional simple waves in elastic bodies subject to internal constraints is presented and it is shown that, with certain qualifications, the aforementioned simplifications continue to hold. Specifically, when n constraints act, the number of simple waves which exist is normally 6–2n, the effect of three or more independent constraints being to suppress one-dimensional simple-wave motion altogether, and explicit integration of the basic equations can be effected for some internal-energy functions when n = 1 and for arbitrary internal energy when n = 2. In addition it is found that the presence of one or two constraints usually admits the possibility that the speed of propagation of the wavelets in a simple wave solution ceases to be real (incompressibility, unaccompanied by any other constraint, being a notable exception in this respect). As illustrations of the theory, simple-wave solutions for three types of single constraints and two examples of double constraints are studied.