In this work, we are concerned with steady-state bifurcation and Turing instability caused by diffusion in two prey-predator models with unbounded exponential or logistic growth in prey population and Allee effect in predator population. We first give stability of the homogeneous steady-state to the model with unbounded exponential growth in prey population, and establish the local structure of the steady states bifurcating from simple and double eigenvalues by the techniques of space decomposition and implicit function theorem. Second, we explore how diffusion destabilizesthe homogeneous steady-state, and explicitly describe the Turing space under certain conditions. Furthermore, some remarks and numerical simulations are illustrated to verify our theoretical findings.