  We present results for exclusive $p p \to p p \pi^+ \pi^-$ or $p \bar p \to p
\bar p \pi^+ \pi^-$ processes at high energies. We discuss the role of new
additional absorption corrections in the non-resonant Lebiedowicz-Szczurek
model mediated by the pomeron and reggeon exchanges. We discuss also the role
of the $\rho^{0}$ and the Drell-S\"oding photoproduction mechanism. We compare
our predictions with recent experimental results obtained by the STAR and CDF
Collaborations. We present predictions for the ALICE, ATLAS and CMS
experiments. Differential distributions in invariant two-pion mass and
two-dimensional distributions in proton-proton relative azimuthal angle and
transverse momentum of one of the protons are presented.
