With one or two exceptions, biological materials are "soft", meaning that they combine viscous and elastic elements. This mechanical behavior results from self-assembled supramolecular structures that are stabilized by noncovalent interactions. It is an ongoing and profound challenge to understand the self-organization of biological materials. In many cases, concepts can be imported from soft-matter physics and chemistry, which have traditionally focused on materials such as colloids, polymers, surfactants, and liquid crystals. Using these ideas, it is possible to gain a new perspective on phenomena as diverse as DNA condensation, protein and peptide fibrillization, lipid partitioning in rafts, vesicle fusion and budding, and others, as discussed in this selective review of recent highlights from the literature.