Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra
A
, how to construct all the Gorenstein-projective
A
-modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra
Λ
=
A
M
B
A
0
B
. We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over
Λ
=
A
M
B
A
0
B
.