Localized surface plasmon resonance (LSPR) in semiconductor nanocrystals (NCs) that results in resonant absorption, scattering, and near field enhancement around the NC can be tuned across a wide optical spectral range from visible to far-infrared by synthetically varying doping level, and post synthetically via chemical oxidation and reduction, photochemical control, and electrochemical control. In this review, we will discuss the fundamental electromagnetic dynamics governing light matter interaction in plasmonic semiconductor NCs and the realization of various distinctive physical properties made possible by the advancement of colloidal synthesis routes to such NCs. Here, we will illustrate how free carrier dielectric properties are induced in various semiconductor materials including metal oxides, metal chalcogenides, metal nitrides, silicon, and other materials. We will highlight the applicability and limitations of the Drude model as applied to semiconductors considering the complex band structures and crystal structures that predominate and quantum effects that emerge at nonclassical sizes. We will also emphasize the impact of dopant hybridization with bands of the host lattice as well as the interplay of shape and crystal structure in determining the LSPR characteristics of semiconductor NCs. To illustrate the discussion regarding both physical and synthetic aspects of LSPR-active NCs, we will focus on metal oxides with substantial consideration also of copper chalcogenide NCs, with select examples drawn from the literature on other doped semiconductor materials. Furthermore, we will discuss the promise that LSPR in doped semiconductor NCs holds for a wide range of applications such as infrared spectroscopy, energy-saving technologies like smart windows and waste heat management, biomedical applications including therapy and imaging, and optical applications like two photon upconversion, enhanced luminesence, and infrared metasurfaces.