A self-contained introduction to the theory of periodic, progressive, permanent waves on the surface of incompressible inviscid fluid. The problem of permanent water-waves has attracted a large number of physicists and mathematicians since Stokes" pioneering papers appeared in 1847 and 1880. Among many aspects of the problem, the authors focus on periodic progressive waves, which mean waves travelling at a constant speed with no change of shape. As a consequence, everything about standing waves are excluded and solitary waves are studied only partly. However, even for this restricted problem, quite a number of papers and books, in physics and mathematics, have appeared and more will continue to appear, showing the richness of the subject. In fact, there remain many open questions to be answered. The book consists of two parts: numerical experiments and normal form analysis of the bifurcation equations. Prerequisite for reading it is an elementary knowledge of the Euler equations for incompressible inviscid fluid and of bifurcation theory. Readers are also expected to know functional analysis at an elementary level.