In this chapter, we will discuss numerical methods for computing wave scattering in periodic structures in photonics, and surface plasmons of electron density waves at interfaces between metallic materials and dielectrics. Both types of electromagnetic phenomena have many applications in nano-photonics and near-field optics. Firstly, we present the general Bloch theory and Bloch wave expansions for electromagnetic waves in periodic structures. Several numerical methods are discussed that can be used for the calculation of the photonic band structure, including a plane wave method and a calculation of transmission spectra by Fourier methods. Then, a modified Nédélec edge element for periodic structure eigenvalue problems in the frequency domain, and a time-domain finite element with Bloch wave transparent boundary conditions, are presented.