This is a unique book that presents rigorous mathematical results on Fermi–Pasta–Ulam lattices, a field of great interest in nonlinear analysis, nonlinear science, mathematical physics, etc. It considers travelling waves and time periodic oscillations in infinite Fermi–Pasta–Ulam lattices, which are not necessarily spatially homogenous. Similar systems, infinite chains of linearly coupled nonlinear oscillators, are also discussed. The book is self-contained and includes a number of open problems, making it suitable for use in a course for graduate students.