  We devise a {\sl non--perturbative} method, called {\sl Parametric
Perturbation Theory} (PPT), which is alternative to the ordinary perturbation
theory. The method relies on a principle of simplicity for the observable
solutions, which are constrained to be linear in a certain (unphysical)
parameter. The perturbative expansion is carried out in this parameter and not
in the physical coupling (as in ordinary perturbation theory). We provide a
number of nontrivial examples, where our method is capable to resum the
divergent perturbative series, {\sl extract} the leading asymptotic (strong
coupling) behavior and {\sl predict} with high accuracy the coefficients of the
perturbative series. In the case of a zero dimensional field theory we prove
that PPT can be used to provide the imaginary part of the solution, when the
problem is analytically continued to negative couplings. In the case of a
$\phi^4$ lattice model 1+1 and of elastic theory we have shown that the
observables resummed with PPT display a branch point at a finite value of the
coupling, signaling the transition from a stable to a metastable state. We have
also applied the method to the prediction of the virial coefficients for a hard
sphere gas in two and three dimensions; in this example we have also found that
the solution resummed with PPT has a singularity at finite density. Predictions
for the unknown virial coefficients are made.
