  The goal of this monograph is to prove that any solution of the Cauchy
problem for the capillarity-gravity water waves equations, in one space
dimension, with periodic, even in space, initial data of small size $\epsilon$,
is almost globally defined in time on Sobolev spaces, i.e. it exists on a time
interval of length of magnitude $\epsilon^{-N}$ for any $N$, as soon as the
initial data are smooth enough, and 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, our
method is based on a normal forms procedure, in order to eliminate those
contributions to the Sobolev energy that are of lower degree of homogeneity in
the solution. Since the water waves equations are a quasi-linear system, usual
normal forms approaches would face the well known problem of losses of
derivatives in the unbounded transformations. In this monograph, to overcome
such a difficulty, after a paralinearization of the capillarity-gravity water
waves equations, necessary to obtain energy estimates, and thus local existence
of the solutions, we first perform several paradifferential reductions of the
equations to obtain a diagonal system with constant coefficients symbols, up to
smoothing remainders. Then we may 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 look for solutions even in $x$, guarantees a key cancellation
which prevents the growth of the Sobolev norms of the solutions.
