Nonlinear cooperative systems associated to vector fields that are concave or subhomogeneous describe well interconnected dynamics that are of key interest for communication, biological, economical, and neural network applications. For this class of positive systems, we provide conditions that guarantee existence, uniqueness and stability of strictly positive equilibria. These conditions can be formulated directly in terms of the spectral radius of the Jacobian of the system. If control inputs are available, then it is shown how to use state feedback to stabilize an equilibrium point in the interior of the positive orthant.