  We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an
algebraically closed field of characteristic zero. First, we consider some
class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It
contains all tori. Linearizability of the natural rational actions of
flattenable subgroups on the affine space $\An$ is intimately related to
rationality of the invariant fields and, for tori, is equivalent to it. We
prove stable linearizability of these actions and show the existence of
nonlinearizable actions among them. This is applied to exploring maximal tori
in $\mathcal C_n$ and to proving the existence of nonlinearizable, but stably
linearizable elements of infinite order in $\mathcal C_n$ for $n\geqslant 6$.
Then we consider some subgroups $\mathcal J(x_1,..., x_n)$ of $\mathcal C_n$
that we call the rational de Jonqui\`eres subgroups. We prove that every affine
algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is solvable and the group of
its connected components is Abelian. We also prove that every reductive
algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is diagonalizable. Further, we
prove that the natural rational action on $\An$ of any unipotent algebraic
subgroup of $\mathcal J(x_1,..., x_n)$ admits a rational cross-section which is
an affine subspace of $\An$. We show that in this statement "unipotent" cannot
be replaced by "connected solvable".\;This is applied to proving a conjecture
of A. Joseph on the existence of "rational slices" for the coadjoint
representations of finite-dimensional algebraic Lie algebras $\mathfrak g$
under the assumption that the Levi decomposition of $\mathfrak g$ is a direct
product.
