  We obtain reliable $\pi\pi$ scattering amplitudes consistent with
experimental data, both at low and high energies, and fulfilling appropriate
analyticity properties. We do this by first fitting experimental low energy
($s^{1/2}\leq1.42 {\rm GeV}$) phase shifts and inelasticities with expressions
that incorporate analyticity and unitarity. In particular, for the S wave with
isospin~0, we discuss in detail several sets of experimental data. This
provides low energy partial wave amplitudes that summarize the known
experimental information. Then, we impose Regge behaviour as follows from
factorization and experimental data for the imaginary parts of the scattering
amplitudes at higher energy, and check fulfillment of dispersion relations up
to 0.925 GeV. This allows us to improve our fits. The ensuing $\pi\pi$
scattering amplitudes are then shown to verify dispersion relations up to 1.42
GeV, as well as $s - t - u$ crossing sum rules and other consistency
conditions. The improved parametrizations therefore provide a reliable
representation of pion-pion amplitudes with which one can test chiral
perturbation theory calculations, pionium decays, or use as input for
CP-violating $K$ decays. In this respect, we find
$[a_0^{(0)}-a_0^{(2)}]^2=(0.077\pm0.008) M^{-1}_\pi$ and
$\delta_0^{(0)}(m^2_K)-\delta_0^{(2)}(m^2_K)=52.9\pm1.6^{\rm o}$.
