Explanations of the late-time cosmic acceleration within the framework of general relativity are plagued by difficulties. General relativistic models are mostly based on a dark energy field with fine-tuned, unnatural properties. There is a great variety of models, but all share one feature in common – an inability to account for the gravitational properties of the vacuum energy, and a failure to solve the so-called coincidence problem. Two broad alternatives to dark energy have emerged as candidate models: these typically address only the coincidence problem and not the vacuum energy problem. The first is based on general relativity and attempts to describe the acceleration as an effect of inhomogeneity in the universe. If this alternative could be shown to work, then it would provide a dramatic resolution of the coincidence problem; however, a convincing demonstration of viability has not yet emerged. The second alternative is based on infra-red modifications to general relativity, leading to a weakening of gravity on the largest scales and thus to acceleration. Most examples investigated so far are scalar-tensor or brane-world models, and we focus on the simplest candidates of each type: f(R) models and DGP models respectively. Both of these provide a new angle on the problem, but they also face serious difficulties. However, investigation of these models does lead to valuable insights into the properties of gravity and structure formation, and it also leads to new strategies for testing the validity of General Relativity itself on cosmological scales.