We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities
C
2
/
G
\mathbb {C}^2/G
where
G
=
1
r
(
1
,
a
)
≤
G
L
(
2
,
C
)
G=\frac {1}{r}(1,a)\leq GL(2,\mathbb {C})
.