The present status and development of strong-field nano-optics, an emerging field of nonlinear optics, is discussed. A nonperturbative regime of light-matter interactions is reached when the amplitude of the external electromagnetic fields that are driving a material approach or exceed the field strengths that bind the electrons inside the medium. In this strong-field regime, light-matter interactions depend on the amplitude and phase of the field, rather than its intensity, as in more conventional perturbative nonlinear optics. Traditionally such strong-field interactions have been intensely investigated in atomic and molecular systems, and this has resulted in the generation of high-harmonic radiation and laid the foundations for contemporary attosecond science. Over the past decade, however, a new field of research has emerged, the study of strong-field interactions in solid-state nanostructures. By using nanostructures, specifically those made out of metals, external electromagnetic fields can be localized on length scales of just a few nanometers, resulting in signficantly enhanced field amplitudes that can exceed those of the external field by orders of magnitude in the vicinity of the nanostructures. This leads not only to dramatic enhancements of perturbative nonlinear optical effects but also to significantly increased photoelectron yields. It resulted in a wealth of new phenomena in laser-solid interactions that have been discovered in recent years. These include the observation of above-threshold photoemission from single nanostructures, effects of the carrier-envelope phase on the photoelectron emission yield from metallic nanostructures, and strong-field acceleration of electrons in optical near fields on subcycle timescales. The current state of the art of this field is reviewed, and several scientific applications that have already emerged from the fundamental discoveries are discussed. These include, among others, the coherent control of localized electromagnetic fields at the surface of solid-state nanostructures and of free-electron wave packets by such optical near fields, resulting in the creation of attosecond electron bunches, the coherent control of photocurrents on nanometer length and femtosecond timescales by the electric field of a laser pulse, and the development of new types of ultrafast electron microscopes with unprecedented spatial, temporal, and energy resolution. The review concludes by highlighting possible future developments, discussing emerging topics in photoemission and potential strong-field nanophotonic devices, and giving perspectives for coherent ultrafast microscopy techniques. More generally, it is shown that the synergy between ultrafast science, plasmonics, and strong-field physics holds promise for pioneering scientific discoveries in the upcoming years. (Less)