Crofton's formula (1867), also known as the Cauchy-Crofton formula, is a beautiful elementary result showing that the length of a curve in IR^2 can be measured by checking how many times a `random' line intersects it. One can think of it as a generalization of Buffon's needle problem and it is the origin of the field of Integral Geometry. We will discuss and prove Crofton's formula and show how it can be used to give short and simple proofs of some nice geometric statements. After that, we will discuss some more recent ideas concerning `bilinear' variants, discuss a new property of convex sets and and introduce the problem of finding sets that `see themselves as little as possible' or, probabilistically, sets that minimize the variance of an intersection with a random line.