In this talk we discuss the dynamics of one-dimensional lattices with nearest neighbor interactions, initially at rest, and which are driven from one end by a particle . The driver, , is assumed to undergo a prescribed motion, are real constants and has period . We describe the numerically observed behaviour (shock and rarefaction phenomena for , generation of multi-phase travelling waves for ) and present corresponding analytical results. Special emphasis is given to the integrable model (Toda lattice). In this particular case one can rigorously derive the long-time asymptotics for a large class of initial value problems using the Inverse Scattering Transform (IST) method. Hereby we formulate the IST as a matrix-valued Riemann-Hilbert problem. Such Riemann-Hilbert problems have recently been used to prove asymptotic results in a variety of different fields, such as integrable systems, statistical mechanics, combinatorics and orthogonal polynomials.