For Legendrian and transverse links in the 3-sphere, Ozsvath, Szabo, and Thurston defined combinatorial invariants that reside in grid homology. Known as the GRID invariants, they are effective in distinguishing some transverse knots that have the same classical invariants. In this talk, we describe some recent developments: First, we show that the GRID invariants obstruct decomposable Lagrangian cobordisms; second, we outline a computable generalization via cyclic branched covers. The first result is joint with John Baldwin and Tye Lidman, and the second with Shea Vela-Vick.