  The Cremona group $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is the group of
birational self-maps of $\mathbb{P}^2_\mathbb{C}$. Using the action of
$\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ on the Picard-Manin space of
$\mathbb{P}^2_\mathbb{C}$ we characterize its solvable subgroups. If
$\mathrm{G}\subset\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is solvable, non
abelian, and infinite, then up to finite index: either any element of
$\mathrm{G}$ is of finite order or conjugate to an automorphism of
$\mathbb{P}^2_\mathbb{C}$, or $\mathrm{G}$ preserves a unique fibration that is
rational or elliptic, or $\mathrm{G}$ is, up to conjugacy, a subgroup of the
group generated by one hyperbolic monomial map and the diagonal automorphisms.
  We also give some corollaries.
