This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use $\theta$-schemes with $\theta\in [\hf,1]$. For the proofs of controllability we rely on the strategy introduced in 1995 by G.~Lebeau and L.~Robbiano for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions' Hilbert Uniqueness Method strategy. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove error estimates with respect to the time step for the control functions obtained in these two cases. The theoretical results are illustrated through numerical experimentations.