This paper shows that the solutions to various convex l1 minimization problems are unique if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other l1 models that either minimize f(Ax-b) or impose the constraint f(Ax-b) <= sigma, where f is a strictly convex function. For these models, this paper proves that, given a solution x* and defining I=supp(x*) and s=sign(x*I), x is the unique solution if and only if AI has full column rank and there exists y such that A'Iy=s and |a'iy|<1 for i not in I. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on I. Indeed, it is also necessary, and applies to a variety of other l1 models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.