In this paper we study the space
S
H
(
A
)
S_{H}(A)
of unital completely positive linear maps from a
C
∗
C^{*}
-algebra
A
A
to the algebra
B
(
H
)
B(H)
of continuous linear operators on a complex Hilbert space
H
H
. The state space of
A
A
, in this notation, is
S
C
(
A
)
S_{\mathbb {C}}(A)
. The main focus of our study concerns noncommutative convexity. Specifically, we examine the
C
∗
C^{*}
-extreme points of the
C
∗
C^{*}
-convex space
S
H
(
A
)
S_{H}(A)
. General properties of
C
∗
C^{*}
-extreme points are discussed and a complete description of the set of
C
∗
C^{*}
-extreme points is given in each of the following cases: (i) the cases
S
C
2
(
A
)
S_{{\mathbb {C}}^{2}}(A)
, where
A
A
is arbitrary ; (ii) the cases
S
C
r
(
A
)
S_{{\mathbb {C}}^{r}}(A)
, where
A
A
is commutative; (iii) the cases
S
C
r
(
M
n
)
S_{{\mathbb {C}}^{r}}(M_{n})
, where
M
n
M_{n}
is the
C
∗
C^{*}
-algebra of
n
×
n
n\times n
complex matrices. An analogue of the Krein-Milman theorem will also be established.