I had the lecture 'Nanooptics II' this semester and just finished it with an oral exam. The lecture was interesting and the professor was good at explaining and presenting the topic. My interest in the matter even grew during the last two weeks while studying for the exam. Repeating the whole lecture in short time and connecting all the topics with themselves and previous knowledge was kinda fun (what studying normally isn't ^^ ).
In a nutshell: Light cannot be focused to an infinitely small spot due to it's wave nature. The diameter of the smallest possible spot with conventional optics (lenses) is roughly half the wavelength (for visible light ~200nm). This limits conventional optical microscopes and makes it impossible for them to see smaller structures with visible light (for example: single molecules).
There are several ways to 'trick' the diffraction limit, but the biggest improvement for resolution is probing the near field, which contains exact spacial information, but cannot propagate (evanescent waves). Basically that means, that we have to look at our object from a very close distance (<<λ). The two techniques the lecture concentrated on where a-SNOM and s-SNOM. With resolutions of up to 10nm one can see single molecules and even get information about the optical properties and carrier-densities of the samples.