The doubly-infinite Toda lattice is a completely integrable system that possesses soliton solutions. The evolution equation for the Toda lattice is equivalent to an isospectral deformation of a doubly-infinite Jacobi matrix, and the initial value problem can be solved by the inverse scattering transform (IST) associated with this Jacobi matrix. We will discuss the numerical computation of the IST for the Toda lattice by solving Riemann-Hilbert problems numerically with the use of the nonlinear steepest descent method. The numerical IST allows one to compute the solution of the initial value problem for arbitrary spatial and temporal parameters, in particular, in the long time scales, with uniform accuracy. Time permitting, we will move onto the long-time behavior of solutions for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice using the direct/inverse scattering approach.