We describe ultraresolution microscopy far beyond the classical Abbe diffraction limit of one half wavelength (lambda/2), and also beyond the practical limit (ca. lambda/10) of aperture-based scanning near-field optical microscopy (SNOM). The 'apertureless' SNOM discussed here uses light scattering from a sharp tip (hence scattering-type or s-SNOM) and has no lambda-related resolution limit. Rather, its resolution is approximately equal to the radius a of the probing tip (for commercial tips, a < 20 nm) so that 10 nm is obtained in the visible (lambda/60). A resolution of lambda/500 has been obtained in the mid-infrared at lambda = 10 microm. The advantage of infrared, terahertz and even microwave illumination is that specific excitations can be exploited to yield specific contrast, e.g. the molecular vibration offering a spectroscopic fingerprint to identify chemical composition. S-SNOM can routinely acquire simultaneous amplitude and phase images to obtain information on refractive and absorptive properties. Plasmon- or phonon-resonant materials can be highlighted by their particularly high near-field signal level. Furthermore, s-SNOM can map the characteristic optical eigenfields of small, optically resonant particles. Lastly, we describe theoretical modelling that explains and predicts s-SNOM contrast on the basis of the local dielectric function.