Plasmonic nanostructures strongly localize electric fields on their surfaces via the collective oscillations of conducting electrons under stimulation by incident light at a certain wavelength. Molecules adsorbed onto the surfaces of plasmonic structures experience a strongly enhanced electric field due to the localized surface plasmon resonance (LSPR), which amplifies the Raman scattering signal obtained from these adsorbed molecules. This phenomenon is referred to as surface-enhanced Raman scattering (SERS). Because Raman spectra serve as molecular fingerprints, SERS has been intensively studied for its ability to facilely detect molecules and provide a chemical analysis of a solution. Further enhancements in the Raman intensity and therefore higher sensitivity in SERS-based molecular analysis have been achieved by designing plasmonic nanostructures with a controlled size, shape, composition, and arrangement. This review paper focuses on the current state of the art in the fabrication of SERS-active substrates and their use as chemical and biosensors. Starting with a brief description of the basic principles underlying LSPR and SERS, we discuss three distinct nanofabrication methods, including the bottom-up assembly of nanoparticles, top-down nanolithography, and lithography-free random nanoarray formation. Finally, typical applications of SERS-based sensors are discussed, along with their perspectives and challenges.