The effects of pressure on protein structure and function can vary dramatically depending on the magnitude of the pressure, the reaction mechanism (in the case of enzymes), and the overall balance of forces responsible for maintaining the protein's structure. Interactions between the protein and solvent are also critical in determining the response of a protein to pressure. Pressure has long been recognized as a potential denaturant of proteins, often promoting the disruption of multimeric proteins, but recently examples of pressure-induced stabilization have also been reported. These global effects can be explained in terms of pressure effects on individual molecular interactions within proteins, including hydrophobic, electrostatic, and van der Waals interactions, which can now be studied in greater detail than ever before. However, many uncertainties remain, and thorough descriptions of how proteins respond to pressure remain elusive. This review summarizes basic concepts and new findings related to pressure effects on intra- and intermolecular interactions within proteins and protein complexes, and discusses their implications for protein structure-function relationships under pressure.