  We study automorphism and birational automorphism groups of varieties over
fields of positive characteristic from the point of view of Jordan and
$p$-Jordan property. In particular, we show that the Cremona group of rank $2$
over a field of characteristic $p>0$ is $p$-Jordan, and the birational
automorphism group of an arbitrary geometrically irreducible algebraic surface
is nilpotently $p$-Jordan of class at most $2$. Also, we show that the
automorphism group of a smooth geometrically irreducible projective variety of
non-negative Kodaira dimension is Jordan in the usual sense.
