Actin filaments and microtubules polymerize and depolymerize by adding and removing subunits at polymer ends, and these dynamics drive cytoplasmic organization, cell division, and cell motility. Since Wegner proposed the treadmilling theory for actin in 1976, it has largely been assumed that the chemical state of the bound nucleotide determines the rates of subunit addition and removal. This chemical kinetics view is difficult to reconcile with observations revealing multiple structural states of the polymer that influence polymerization dynamics but that are not strictly coupled to the bound nucleotide state. We refer to these phenomena as "structural plasticity" and discuss emerging evidence that they play a central role in polymer dynamics and function.