  Let $\mathbf{K}$ be an algebraically closed field. The Cremona group
$\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of
the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study
of centralizers of elements of infinite order in
$\operatorname{Cr}_2(\mathbf{K})$ which leads to a classification of embeddings
of $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a
classification of maximal non-torsion abelian subgroups of
$\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.
