  We analyze the Roy equations for the lowest partial waves of elastic pi pi
scattering and demonstrate that the two S-wave scattering lengths a_0^0 and
a_0^2 are the essential parameters in the low energy region: Once these are
known, the available experimental information determines the behaviour near
threshold to within remarkably small uncertainties. An explicit numerical
representation for the energy dependence of the S- and P-waves is given and it
is shown that the threshold parameters of the D- and F-waves are also fixed
very sharply in terms of a_0^0 and a^2_0. In agreement with earlier work, which
is reviewed in some detail, we find that the Roy equations admit physically
acceptable solutions only within a band of the (a_0^0,a_0^2) plane. We show
that the data on the reactions e+e- -> pi pi and tau -> pi pi nu reduce the
width of this band quite significantly. Furthermore, we discuss the relevance
of the decay K -> pi pi e nu in restricting the allowed range of a_0^0,
preparing the grounds for an analysis of the forthcoming precision data on this
decay and on pionic atoms. We expect these to reduce the uncertainties in the
two basic low energy parameters very substantially, so that a meaningful test
of the chiral perturbation theory predictions will become possible.
