The existence of discretely holomorphic observables played a pivotal role in the recent rigorous proof of conformal invariance in the Ising model. In this presentation, we show the close relationship between solvability and discrete holomorphicity for three families of models that are natural generalizations of the Ising model, namely, the Potts models, the O(n) loop models and the Z_N models. It is shown that in all these cases, the holomorphicity of some appropriate observables imply the criteria of integrability, namely, the Yang-Baxter relations and the inversion relations. Furthermore, it is also found that the value of the observable at the boundary of a Baxter lattice, when embedded isoradially onto the complex plane, remains invariant under topological movements of the defining rapidity lines, and thus defines the observable on the shared boundary for the equivalence class of Baxter lattices.