A characterization of tame automorphisms of the algebra
A
=
F
[
x
1
,
x
2
,
x
3
]
A=F[x_1,x_2,x_3]
of polynomials in three variables over a field
F
F
of characteristic
0
0
is obtained. In particular, it is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of
A
A
are algorithmically recognizable.