Computer simulations of proteins, lipids and nucleic acids at equilibrium have become essentially routine. However, the fact remains that complete sampling of conformational space continues to be a bottle-neck in the field. The challenge for the future is to overcome such problems and use computational approaches to understand recognition and spontaneous self-organization in biomolecular systems (folding, aggregation and assembly of complexes), processes that cannot be directly observed experimentally. In this review, examples illustrating the extent to which simulations can be used to understand these phenomena in biomolecular systems will be presented along with examples of methodological developments to increase our physical understanding of the processes. The study cases will cover the problems of peptide-receptor recognition and the use of the information obtained for the design of new non-peptidic ligands; the study of the folding mechanism of small proteins and finally the study of the initial stages of peptide self-aggregation.