This paper reviews the substantial body of literature emerging since 2004 concerning photonic nanojets. The photonic nanojet is a narrow, high-intensity, non-evanescent light beam that can propagate over a distance longer than the wavelength λ after emerging from the shadow-side surface of an illuminated lossless dielectric microcylinder or microsphere of diameter larger than λ. The nanojet's minimum beamwidth can be smaller than the classical diffraction limit, in fact as small as ~λ/3 for microspheres. It is a nonresonant phenomenon appearing for a wide range of diameters of the microcylinder or microsphere if the refractive index contrast relative to the background is less than about 2:1. Importantly, inserting within a nanojet a nanoparticle of diameter d(ν) perturbs the far-field backscattered power of the illuminated microsphere by an amount that varies as d(ν)3 for a fixed λ. This perturbation is much slower than the d(ν)6 dependence of Rayleigh scattering for the same nanoparticle, if isolated. This leads to a situation where, for example, the measured far-field backscattered power of a 3-μm diameter microsphere could double if a 30-nm diameter nanoparticle were inserted into the nanojet emerging from the microsphere, despite the nanoparticle having only 1/10,000(th) the cross-section area of the microsphere. In effect, the nanojet serves to project the presence of the nanoparticle to the far field. These properties combine to afford potentially important applications of photonic nanojets for detecting and manipulating nanoscale objects, subdiffraction-resolution nanopatterning and nanolithography, low-loss waveguiding, and ultrahigh-density optical storage.