Bihamiltonian structures (pairs of compatible Hamiltonian operators) play an important role in the theory of integrability of systems of evolutionary PDEs. Well through the 1980's, a dominant problem in the theory was the classification of bihamiltonian structures with prescribed highest order terms. In the early 1990's, B. Dubrovin proposed fixing the zeroth and first order terms of the bihamiltonian structures and considering formal deformations thereof. In this talk, we will explain how the cohomology governing these deformations is that of a certain double complex, and present recent results on their classifications and the algebraic structures underlying them. Time permitting, we also will discuss how bihamiltonian hierarchies corresponding to such deformations arise in the context of Frobenius manifolds and the study of 2-dimensional topological field theory.