Two approaches for the adaptive predictive control of SISO linear time-invariant plants subject to saturations on both the control variable and its increment are investigated. In the desaturating approach, the saturations are disregarded in the design phase, and their impact on performance is reduced by resorting to an anti-windup implementation of the controller. In the optimisation approach, at each time step an optimisation problem is solved that explicitly takes into account the presence of saturation constraints. Using a predictive control scheme with terminal input and output constraints, BIBO stability is proven for both approaches in the non-adaptive case. These results are exploited to prove BIBO stability also in the adaptive case for the desaturating approach as well as for a mixed desaturation/optimisation scheme.