Graphene has shown much promise as an organic electronic material but, despite recent achievements in the production of few-layer graphene, the quantitative exfoliation of graphite into pristine single-layer graphene has remained one of the main challenges in developing practical devices. Here we report that microwave irradiation of graphite suspended in molecularly engineered oligomeric ionic liquids allows for ultrahigh-efficiency exfoliation (93% yield) with a high selectivity (95%) towards ‘single-layer’ graphene (that is, with thicknesses <1 nm) in a short processing time (30 minutes). The isolated graphene sheets show negligible structural deterioration. They are also readily redispersible in oligomeric ionic liquids up to ∼100 mg/ml, and form physical gels in which an anisotropic orientation of graphene sheets, once induced by a magnetic field, is maintained.