We discuss the history of the four famous problems of antiquity: squaring the circle, trisecting the angle, doubling the cube, and constructing regular n-gons. There is a long and fascinating history of mathematicians´ failures to solve the problems using the compass and straightedge and their successes at solving them by other means (marked straightedges, conic sections, transcendental curves, mechanical devices, etc.). Like all great mathematical problems they pushed mathematics forward, culminating in the proofs that they are impossible to solve using the Euclidean tools.