Roughly speaking, a conformal space is a differentiable manifold
M
n
{M^n}
in which the notion of angle of tangent vectors at a point
p
∈
M
n
p \in {M^n}
makes sense and varies differentiably with
p
p
; two such spaces are (locally) equivalent if they are related by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space
R
n
{R^n}
. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure; in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, assume the ambient space to be
R
n
+
1
{R^{n + 1}}
. Then, if
n
⩾
4
n \geqslant 4
, a conformally flat hypersurface
M
n
⊂
R
n
+
1
{M^n} \subset {R^{n + 1}}
can be described as follows. Diffeomorphically,
M
n
{M^n}
is a sphere
S
n
{S^n}
with
b
1
(
M
)
{b_1}(M)
handles attached, where
b
1
(
M
)
{b_1}(M)
is the first Betti number of
M
M
. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of
R
n
+
1
{R^{n + 1}}
that are foliated by complete round
(
n
−
1
)
(n - 1)
-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of
R
n
+
1
{R^{n + 1}}
: (a) an open piece of an
n
n
-sphere or an
n
n
-plane bounded by round
(
n
−
1
)
(n - 1)
-sphere, (b) a round
(
n
−
1
)
(n - 1)
-sphere, (c) a point.