The slide polynomials form a basis of the polynomial ring that lifts I. Gessel's fundamental basis of quasisymmetric functions. We review recent results concerning the slide basis, including the fact that it has positive structure constants, and that elements of the geometrically-important Schubert and key polynomial bases expand positively in the slide basis. We also discuss properties of the glide basis of the polynomial ring. Glide polynomials are K-theoretic analogues of slide polynomials: they refine Grothendieck polynomials (which represent K-theory Schubert classes for the flag variety) in the same way slide polynomials refine Schubert polynomials. This talk contains joint work with S. Assaf and (separately) O. Pechenik.