The nature of dark energy, responsible for the observed accelerated expansion of our universe, is one of the biggest puzzles in contemporary physics. String theory, as a candidate fundamental theory, would provide an origin to dark energy in the form of an effective scalar potential V. Getting the right shape for V to match observational constraints is however challenging, and this has led to an important recent activity. In this talk, I will first review the status of classical de Sitter solutions (critical points of V that would give a positive cosmological constant). I will then present possibilities for an asymptotic accelerated expansion, meaning realising dark energy by a field rolling in V towards the asymptotics. In this context, allowing for an open universe (i.e. a negatively curved space) offers important possibilities to string theory cosmological models.