Nuclear fuel claddings are made of zirconium alloys and because of their exposure to water (coolant) during the operation and throughout the storage period, zirconium-oxidation (zirconia), phase transformation and hydride precipitates occur in the cladding material which may result in the failure of nuclear fuel rods. In this work, different phase field models are presented to capture oxidation of zirconium, phase transformation in zirconia, and γ-hydride precipitation in nuclear fuel claddings. Evolutions of non-conserved phase field variables (e.g., phase transformation variants) are simulated by Ginzburg-Landau time-dependent equations, and the evolution of conserved phase field variables (e.g., concentration of hydrogen) are simulated by Cahn-Hilliard time-dependent equations. The elastic strain energy generated by oxidation, phase transformation or γ-hydride precipitation is added to the total free energy functional of each phase field model, and the mechanical equilibrium equations are solved at each time step of the simulations. For each model, coupled evolution equations of phase field variables and the mechanical equilibrium equations are solved in a finite element frame work. Resulting models are capable of predicting composition depth and stress profiles in the zirconium oxide layer, tetragonal-to-monoclinic phase transformation, shape memory effect and pseudoelasticity in zirconia, and γ-hydride precipitation and resulting stresses in zirconium cladding material.