Optical microscopy with nanoscale resolution, beyond that which is possible with conventional diffraction–limited microscopy, may be achieved by scanning a nanoantenna in close proximity to a sample surface. This review will first aim to provide an overview of the basic principles of this technique of scanning near–field optical microscopy (SNOM), before moving on to consider the most widely implemented form of this microscopy, in which the sample is illuminated through a small aperture held less than 10 nm from the sample surface for optical imaging with a resolution of ca. 50 nm. As an example of the application of this microscopy, the results of SNOM measurements of light–emitting polymer nanostructures are presented. In particular, SNOM enables the unambiguous identification of the different phases present in the nanostructures, through the local analysis of the fluorescence from the polymers. The exciting new possibilities for high–resolution optical microscopy and spectroscopy promised by apertureless SNOM techniques are also considered. Apertureless SNOM may involve local scattering of light from a sample surface by a tip, local enhancement of an optical signal by a metal tip, or the use of a fluorescent molecule or nanoparticle attached to a tip as a local optical probe of a surface. These new optical nanoprobes offer the promise of optical microscopy with true nanometre spatial resolution.