  (Due to the limit on the number of characters for an abstract set by arXiv,
the full abstract can not be displayed here. See the abstract in the paper.) We
study the construction of the $\Phi^3_3$-measure and complete the program on
the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz,
Rose, and Speer (1988). This problem turns out to be critical, exhibiting the
following phase transition. In the weakly nonlinear regime, we prove
normalizability of the $\Phi^3_3$-measure and show that it is singular with
respect to the massive Gaussian free field. Moreover, we show that there exists
a shifted measure with respect to which the $\Phi^3_3$-measure is absolutely
continuous. In the strongly nonlinear regime, by further developing the
machinery introduced by the authors (2020), we establish non-normalizability of
the $\Phi^3_3$-measure. Due to the singularity of the $\Phi^3_3$-measure with
respect to the massive Gaussian free field, this non-normalizability part poses
a particular challenge as compared to our previous works. In order to overcome
this issue, we first construct a $\sigma$-finite version of the
$\Phi^3_3$-measure and show that this measure is not normalizable. Furthermore,
we prove that the truncated $\Phi^3_3$-measures have no weak limit in a natural
space, even up to a subsequence.
  We also study the dynamical problem. By adapting the paracontrolled approach,
in particular from the works by Gubinelli, Koch, and the first author (2018)
and by the authors (2020), we prove almost sure global well-posedness of the
hyperbolic $\Phi^3_3$-model and invariance of the Gibbs measure in the weakly
nonlinear regime. In the globalization part, we introduce a new, conceptually
simple and straightforward approach, where we directly work with the
(truncated) Gibbs measure, using the variational formula and ideas from theory
of optimal transport.
