We are used to dividing nature into a macroscopic and a microscopic world. The macroscopic world contains the things we can see with our eyes. The microscopic world contains the building blocks of matter, the atoms and molecules. We know they are there, but we can't see them directly. The mesoscopic world is in between the microscopic and the macroscopic world. The boundaries are not sharp, but can be roughly indicated. Mesoscopic and macroscopic objects have in common that they both contain a large number of atoms. A first difference is that the macroscopic object can be well described by the average properties of the material from which it is made. The mesoscopic object, in contrast, is so small that fluctuations around the average become important. A second difference is that the macroscopic object obeys (to a good approximation) the laws of classical mechanics, whereas the mesoscopic object is so small that these laws no longer hold. Mesoscopic and microscopic systems both belong to the wonderful world of quantum mechanics.
Mesoscopic physics addresses fundamental physical problems which occur when a macroscopic object is miniaturized. The field originated some ten years ago, motivated largely by the electronics industry. As you know, that industry makes money out of the miniaturization of transistors, which switch the electrical current on a computer chip. I was fortunate enough to work at the Philips Research Labs. when the field was just starting, and have been deeply involved in it ever since.