Given two independent Poisson point processes Phi(1),Phi(2) in Rd the AB Poisson Boolean model is the graph with points of Phi(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centred at these points contains at least one point of Phi(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all $d > 1$ and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and cn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest neighbour distance and almost sure asymptotic bounds for the connectivity threshold.