Nanostructures made of dielectric materials with high or moderate refractive indexes can support strong electric and magnetic resonances in the optical region. They can therefore function as nanoresonators. In addition to plasmonic metal nanostructures that have been widely investigated, dielectric nanoresonators provide a new type of building blocks for realizing powerful and versatile nanoscale light manipulation. In contrast to plasmonic metal nanostructures, nanoresonators made of appropriate dielectric materials are low-cost, earth-abundant and have very small or even negligible light energy losses. As a result, they will find potential applications in a number of photonic devices, especially those that require low energy losses. In this review, we describe the recent progress on the experimental and theoretical studies of dielectric nanoresonators. We start from the basic theory of the electromagnetic responses of dielectric nanoresonators and their fabrication methods. The optical properties of individual dielectric nanoresonators are then elaborated, followed by the coupling behaviors between dielectric nanoresonators, between dielectric nanoresonators and substrates, and between dielectric nanoresonators and plasmonic metal nanostructures. The applications of dielectric nanoresonators are further described. Finally, the challenges and opportunities in this field are discussed.