The goal of this monograph is to prove that any solution of the Cauchy problem for the
capillary-gravity water waves equations in one space dimension with periodic even in space
small and smooth enough initial data is almost globally defined in time on Sobolev spaces
provided the gravity-capillarity parameters are taken outside an exceptional subset of zero
measure. In contrast to the many results known for these equations on the real line with
decaying Cauchy data one cannot make use of dispersive properties of the linear flow. Instead
a normal forms-based procedure is used eliminating those contributions to the Sobolev energy
that are of lower degree of homogeneity in the solution. Since the water waves equations form a
quasi-linear system the usual normal forms approaches would face the well-known problem of
losses of derivatives in the unbounded transformations. To overcome this after a
paralinearization of the capillary-gravity water waves equations we perform several
paradifferential reductions to obtain a diagonal system with constant coefficient symbols up
to smoothing remainders. Then we start with a normal form procedure where the small divisors
are compensated by the previous paradifferential regularization. The reversible structure of
the water waves equations and the fact that we seek solutions even in space guarantees a key
cancellation which prevents the growth of the Sobolev norms of the solutions.
