The unique electrical, optical, and chemical properties of the two-dimensional crystal of carbon, graphene, stimulated great interest for plasmonics in reduced dimensions. Notably, controlled electrostatic doping on graphene opens new perspectives for gate-tunable active plasmonics. In this chapter, we review the fundamental aspects of plasmons on graphene and their applications ranging from surface plasmon sensors to active plasmonic devices. 11.1 Introduction: Plasmons in Reduced DimensionsGraphene is a two-dimensional (2D) crystal. Studying plasma oscillation on 2D materials sometimes causes confusions between surface plasmons and surface plasmon polaritons (SPPs). First, we would like to clarify these definitions. Plasmons are quasiparticles of quantized plasma oscillations. Polaritons are also quasiparticles, but they are formed by strong coupling of electromagnetic waves and excitations on a material. For example, (SPPs) are the coupled state of plasmons and electromagnetic waves that propagate on the surface of 3D material. Plasmons on 2D materials are usually named as surface plasmons because they live on a surface of 2D material but they are not polaritons. When we say surface plasmons of graphene we talk about bulk plasma oscillations on graphene surface. To understand the physics of plasmons, we should start with the comparison of plasma oscillations in different dimensions such as 3D bulk materials, i.e., gold and silver; 2D materials, i.e., graphene and inversion layer on silicon; 1D materials, i.e., carbon nanotubes or atomic nanowires. Depending on the dimensionality of the material, dispersion characteristics of the plasma oscillations differ. For 3D materials, the plasma oscillations are dispersionless with a plasma frequency of 23D 4= nempw , (11.1)where e is the elementary charge, m is the mass of an electron, and n is the charge density, which is the only material-dependent parameter that defines the plasma frequency. In reduced dimensions, however, plasma oscillations yield dispersion characteristics because the electric field of the plasma oscillations penetrates out of the material resulting in less restoring forces. This causes momentum-dependent plasma frequency. The early experiments on 2D plasma oscillations have been performed on liquid helium surface by Gregory Adams in 1976 . The image-potential-induced surface states on liquid helium traps electrons in close proximity to the surfaces. These trapped electrons form a sheet of 2D electron gas with extremely large carrier mobilities of 107 cm2/Vs. However, the charge densities on liquid helium are limited to 108 cm-2. Later, metal-oxide-semiconductor (MOS) structures have been used to generate tunable 2D electron gas in the inversion layer on silicon surface . A grating-shaped gate electrode was used to generate the inversion layer and to couple light to plasmons. With Si-MOS devices, carrier densities of 4 × 1012 cm-2 with plasma frequency of 30 cm-1 can be achieved. Very recently, graphene has been used to reveal the physics of 2D-plasmons [4, 5]. Graphene in back- gatedtransistor geometry provides a unique system to study plasma oscillations. Various techniques have been used to probe the plasmons on graphene sheet. Because of large momentum mismatch, plasmon on graphene cannot be excited by free propagating light. Therefore, a sharp atomic force microscope tip attached with an infrared spectrometer has been used to probe the tunable graphene plasmon in a back-gated transistor.