  We classify up to conjugacy the subgroups of certain types in the full, in
the affine, and in the special affine Cremona groups. We prove that the
normalizers of these subgroups are algebraic. As an application, we obtain new
results in the Linearization Problem generalizing to disconnected groups
Bia\lynicki-Birula's results of 1966-67. We prove "fusion theorems" for
$n$-dimensional tori in the affine and in the special affine Cremona groups of
rank $n$. In the final section we introduce and discuss the notions of Jordan
decomposition and torsion primes for the Cremona groups.
