In this project, I used various numerical techniques to investigate the existence of nonlinear, spatially localized modes for vibrational and magnetic lattices. I started by using Newton-Raphson algorithm to solve the well-known 1-D Fermi-Pasta-Ulam (FPU) lattice for nonlinear, stable and localized solutions, also known as intrinsic localized modes (ILMs). Using this approach, I found lattice ILMs at the nonlinear frequency just above the minimum nonlinear threshold, then input them into Newton-Raphson to obtain more solutions at higher frequencies via continuation. I then evolved these solutions in time with Runge-Kutta (RK4) algorithm to evaluate their stability. Afterwards, I turned to the 1-D ferromagnetic and antiferromagnetic lattices, and tried a similar approach to the FPU lattice. Due to the huge in- crease in complexity of the magnetic lattices and possibly other complications, the Newton-Raphson algorithm failed to converge even when backtracking line search and trust-region constraint variations were implemented. Thus I used the shooting method to solve for ILMs in the lattice, and successfully found ILMs in the antiferromagnetic lattice. Afterwards, I also explored both the formation of nontrivial localized modes in 1-D and 2-D ferromagnetic lattice from small perturbations from the uniform mode and their long term stability. ii