  Conformal blocks are the building blocks for correlation functions in
conformal field theories. The four-point function is the most well-studied
case. We consider conformal blocks for $n$-point correlation functions. For
conformal field theories in dimensions $d=1$ and $d=2$, we use the shadow
formalism to compute $n$-point conformal blocks, for arbitrary $n$, in a
particular channel which we refer to as the comb channel. The result is
expressed in terms of a multivariable hypergeometric function, for which we
give series, differential, and integral representations. In general dimension
$d$ we derive the $5$-point conformal block, for external and exchanged scalar
operators.
