The existence, uniqueness and multiplicity of positive solutions to a diffusive predator-prey model with B-D functional response and Allee effect are discussed. By the fixed point index theory, the sufficient conditions for the existence of positive solutions are obtained. Secondly, the conditions for the uniqueness of positive solutions are given by the variational characterization of the lowest eigenvalue. Finally, based on the analysis of positive solutions to two limiting systems, the exact multiplicity and stability of positive solutions are determined by means of the combination of the fixed point index theory, bifurcation theory and perturbation theory of eigenvalues. When the Allee effect constants meet appropriate relationship and the growth rate of the predator is large, the results show that the system has only a unique positive solution when the parameters satisfy certain conditions, and has exactly two positive solutions when the growth rate of the prey lies in a certain range.