We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is either a conditional Gaussian vector or a conditional Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for an other probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Solé, Taqqu & Utzet [31] for Gaussian approximation; and by Peccati [32] for Poisson approximations, as well as the recently established fourth-moment theorems on the Poisson space of Döbler & Peccati [7]. Applications to stochastic processes are given.