Directed polymer models are finite-temperature versions of first- and last-passage percolation on the lattice. In 1+1 dimensions, the free-energy of the directed polymer is conjecturally in the Tracy-Widom universality class at all finite temperatures. Tracy-Widom universality has only been proven for a small class of polymers – the so-called solvable models that include Seppalainen's gamma polymers and the O'Connell-Yor semi-discrete polymer - with very special shapes and edge-weight distributions. We present some new fluctuation results towards the universality conjecture for polymers in the intermediate disorder scaling regime. (joint work with Jeremy Quastel)