Brillouin zones and geometry of numbers

It is shown that the Brillouin zones (which originally appeared in the quantum solid state theory) have a series of outstanding purely geometrical and arithmetical properties. In their terms some problems of geometry of numbers concerning the best packing and covering of circles and balls by fundamental domains of a lattice can be solved. Also it turns out that geometry of the Brillouin zones is closely connected with the classical numbertheoretic problems concerning the number of lattice points inside a circle and a ball.