We first consider the problem of finding all the entire area-stationary graphs in the sub-Riemannian Heisenberg group $H_1$.This question has been well studied for t-graphs and for intrinsicgraphs. In particular, examples of unstable entire area-stationary graphs may be given in $H_1$. After discussing some stability results for complete graphs we state a joint result with A. Hurtado and M. Ritoré, where we prove that a $C^2$ complete stable area-stationary surface in $H_1$ must be a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.