Reaching light intensities above 1025 W cm−2 and up to the Schwinger limit of order 1029 W cm−2 would enable the testing of fundamental predictions of quantum electrodynamics. A promising—yet challenging—approach to achieve such extreme fields consists in reflecting a high-power femtosecond laser pulse off a curved relativistic mirror. This enhances the intensity of the reflected beam by simultaneously compressing it in time down to the attosecond range, and focusing it to submicrometre focal spots. Here we show that such curved relativistic mirrors can be produced when an ultra-intense laser pulse ionizes a solid target and creates a dense plasma that specularly reflects the incident light. This is evidenced by measuring the temporal and spatial effects induced on the reflected beam by this so-called plasma mirror. The all-optical measurement technique demonstrated here will be instrumental for the use of relativistic plasma mirrors with the upcoming generation of petawatt lasers that recently reached intensities of 5 × 1022 W cm−2, and therefore constitutes a viable experimental path to the Schwinger limit.