A balanced complex of type
(
a
1
,
…
,
a
m
)
({a_1},\ldots ,{a_m})
is a finite pure simplicial complex
Δ
\Delta
together with an ordered partition
(
V
1
,
…
,
V
m
)
({V_1},\ldots ,{V_m})
of the vertices of
Δ
\Delta
such that card
(
V
i
∩
F
)
=
a
i
({V_i}\, \cap \,F)\, = \,{a_i}
, for every maximal face F of
Δ
\Delta
. If
b
=
(
b
1
,
…
,
b
m
)
{\mathbf {b}}\, = \,({b_1},\ldots ,{b_m})
, then define
f
b
(
Δ
)
{f_\textbf {b}}(\Delta )
to be the number of
F
∈
Δ
F\, \in \,\Delta
satisfying card
(
V
i
∩
F
)
=
b
i
({V_i}\, \cap \,F)\, = \,{b_i}
. The formal properties of the numbers
f
b
(
Δ
)
{f_\textbf {b}}(\Delta )
are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers
f
b
(
Δ
)
{f_\textbf {b}}(\Delta )
. For a certain complex
Δ
(
P
)
\Delta (P)
coming from a poset P, our results are intimately related to properties of the Möbius function of P.