The growing interest in the development of new platforms for the application of Raman spectroscopy techniques in biosensor technologies is driven by the potential of these techniques in identifying chemical compounds, as well as structural and functional features of biomolecules. The effect of Raman scattering is a result of inelastic light scattering processes, which lead to the emission of scattered light with a different frequency associated with molecular vibrations of the identified molecule. Spontaneous Raman scattering is usually weak, resulting in complexities with the separation of weak inelastically scattered light and intense Rayleigh scattering. These limitations have led to the development of various techniques for enhancing Raman scattering, including resonance Raman spectroscopy (RRS) and nonlinear Raman spectroscopy (coherent anti-Stokes Raman spectroscopy and stimulated Raman spectroscopy). Furthermore, the discovery of the phenomenon of enhanced Raman scattering near metallic nanostructures gave impetus to the development of the surface-enhanced Raman spectroscopy (SERS) as well as its combination with resonance Raman spectroscopy and nonlinear Raman spectroscopic techniques. The combination of nonlinear and resonant optical effects with metal substrates or nanoparticles can be used to increase speed, spatial resolution, and signal amplification in Raman spectroscopy, making these techniques promising for the analysis and characterization of biological samples. This review provides the main provisions of the listed Raman techniques and the advantages and limitations present when applied to life sciences research. The recent advances in SERS and SERS-combined techniques are summarized, such as SERRS, SE-CARS, and SE-SRS for bioimaging and the biosensing of molecules, which form the basis for potential future applications of these techniques in biosensor technology. In addition, an overview is given of the main tools for success in the development of biosensors based on Raman spectroscopy techniques, which can be achieved by choosing one or a combination of the following approaches: (i) fabrication of a reproducible SERS substrate, (ii) synthesis of the SERS nanotag, and (iii) implementation of new platforms for on-site testing.