The study of Localized solutions has been a classic topic in nonlinear science, which for instance encourages the advance of the theories for solitary waves. On the other hand,  recent developments in topological materials (such as topological insulators) provide a large number of examples and applications for linear localized states (e.g., edge states). In this talk, we will give an introduction to linear and nonlinear localized states in lattices, with emphasis on attention-attracting edge states. Among other open questions on nonlinear lattices, we will present some new results on the finite-size effects and use a simple diatomic chain model to show how these are relevant to a typical formation mechanism for nonlinear localized states. Here we introduce the Large-Lattice limit to provide a framework for conducting asymptotic analysis in finite-size lattices, which is expected to be extended in a wider range of lattice systems.