** Abstract: **
We study algebraic group actions on central simple algebras.
Suppose and algebraic group G acts on a central simple algebra A
of degree n. We are interested in questions of the following type:

(a) Do the G-fixed elements form a central simple subalgebra of A
of degree n?
(b) Does A have a G-invariant maximal subfield?
(c) Does A have a splitting field
with a G-action, extending the G-action on the center of A?

Somewhat surprisingly, we find that under mild
assumptions on A and the actions, one can answer these questions
by using techniques from birational invariant theory
(i.e., the study of group actions on algebraic varieties, up to
an equivariant birational isomorphism). In fact, group actions
on central simple algebras
turn out to be related to some of the central problems in
birational invariant theory, such as the existence of sections,
stabilizers in general position, affine models, etc. In this paper
we explain these connections and explore them to give partial
answers to questions (a) - (c).