  We prove that any group of cardinality at most the one of $\mathbb{C}$ is a
quotient of any Cremona group of rank at least $4$. This provides a definitive
answer to the question of what the quotients of Cremona groups can be. As a
consequence, this gives a negative answer to the question of I. Dolgachev of
whether Cremona groups of all ranks are generated by involutions. As another
application, we show that higher Cremona groups do not enjoy some classical
group-theoretic properties (namely, the Hopfian property) which are satisfied
by Cremona groups of rank $2$. Finally, we discover that the $3$-torsion of the
Cremona group of rank at least $4$ is not countable.
  To deduce these properties of higher Cremona groups, we first describe the
group of birational transformations of a non-trivial Severi-Brauer surface $S$
over a perfect field, proving in particular that if $S$ contains a point of
degree $6$, then its group of birational self-maps is not generated by elements
of finite order as it admits a surjective group homomorphism to~$\mathbb{Z}$.
We then use this result to study Mori fibre spaces over the field of complex
numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.
