The optical excitation of surface plasmons in metal nanoparticles leads to nanoscale spatial confinement of electromagnetic fields. The confined electromagnetic fields can generate intense, localized thermal energy and large near-field optical forces. The interaction between these effects and nearby molecules has led to the emerging field known as molecular plasmonics. Recent advances in molecular plasmonics have enabled novel optical materials and devices with applications in biology and nanomedicine. In this article, we categorize three main types of interactions between molecules and surface plasmons: optical, thermal and mechanical. Within the scope of each type of interaction, we will review applications of molecular plasmonics in biology and nanomedicine. We include a wide range of applications that involve sensing, spectral analysis, imaging, delivery, manipulation and heating of molecules, biomolecules or cells using plasmonic effects. We also briefly describe the physical principles of molecular plasmonics and progress in the nanofabrication, surface functionalization and bioconjugation of metal nanoparticles.