**1. Introduction**

Almost 50 years ago, in 1974, Fleischmann et al. observed an enhanced Raman signal from a monolayer coverage of pyridine molecules adsorbed on a silver electrode [1]. Such unexpected behavior opened and initiated a new field of spectroscopic analysis including experimental and theoretical studies. Creighton's group [2] and Van Duyne's group [3] published similar results regarding pyridine molecules on silver anode surfaces using relatively low laser power and they brought forward different explanations for their results on the variation of Raman signal. Creighton explained that the enhanced signal is, on one hand, due to a surface effect increasing the molecular Raman scattering cross section and, on the other hand, due to the broadening of the electronic energy levels of molecules at rough metal surfaces, which may induce resonant Raman scattering from the adsorbed molecules *via* interaction with surface plasmons. Additionally, Van Duyne discovered that the enhancement of the

Raman signal is related to the roughness of the surface of the silver electrode and together they established the hypothesis that the phenomenon is due to electric field enhancement. Both explanations were accepted, but the electromagnetic enhancement was more prevailing [4] compared with the chemical or first-layer enhancement when molecules are in contact with metals.

In 1978, Moskovitz [5] put forward a possible explanation for the excitationwavelength-dependent behavior observed by Creighton et al. [6] and Jeanmarie and Van Duyne [3]. He suggested that the observed behavior originates from the adsorbate-covered metal bumps on the metal surface, which could be considered as a two-dimensional colloid of metal spheres covered with the adsorbate and embedded in a dielectric medium on top of a smooth metal mirror. Additional absorption appears for those metal colloidal particles, for which interband electronic transitions exist [7].

Such resonance from the collective oscillation of conduction electrons on the metal surface and limited by the physical dimensions of the metal nanostructure is called localized surface plasmon resonance (LSPR). Typical metals used for plasmonic nanostructures are copper, silver, and gold [8, 9]. Due to the strong confinement of the electric field in the vicinity of such plasmonic metal nanostructures, they can be employed to break the optical diffraction limit and offer a vast range of applications in the fields of biology [10], chemistry [11–14], information [15], optical devices [16–19], and energy science [20, 21]. As a fast-developing field, researchers have studied plasmonic-related phenomena in all directions including the ultimate theoretical understanding from classical theory (e.g., Mie theory [22–26]) to quantum theory [27–29] aiming at providing a solid theoretical background to the main experimental and technological applications.

It is known that many factors, such as the material, size, shape, and dielectric environment, play important roles in determining the LSPR [30]. Among these factors, the geometry [31–37] of the plasmonic nanostructure provides the largest freedom and a straightforward way to tune the plasmonic resonance condition. Therefore, in this chapter, we focus on different geometries of plasmonic structures starting from introducing mathematical solutions for a single metal sphere representing a monomer system and two coupled metal spheres representing a dimer system. Then, we give a review on theoretical approaches that have been used in two powerful analytical techniques: SERS and TERS. Both experiment and simulation provide solid input to the understanding of the mechanisms and the principles of the techniques.
