In the 1980's, knotting in DNA became a fundamental research dimension in the study of the mechanisms by which enzymes act on it. Later, the first compelling identification of knotting in proteins, in 2000, launched the study of knotting in protein structures, and linear macromolecules more generally, following on theoretical efforts of the 1960's. The linking occurring in structures such as DNA, with the articulation of the relationship between linking, twisting, and writhe, and, more directly, linking in Olympic gels has been of interest to geometers, molecular biologists, and polymer physicists since the 1960's. More recently, a new mathematical analysis of both global and local facets of knotting and linking is providing promising discoveries. Following a discussion of the topological structures of knotting and linking, we will consider some of their applications, and close with a consideration of new questions that suggest attractive directions for future research.