  In this paper we utilize $\zeta$-function regularization techniques in order
to compute the Casimir force for massless scalar fields subject to Dirichlet
and Neumann boundary conditions in the setting of the conical piston. The
piston geometry is obtained by dividing the bounded generalized cone into two
regions separated by its cross section positioned at $a$ with $a\in(0,b)$ with
$b>0$. We obtain expressions for the Casimir force that are valid in any
dimension for both Dirichlet and Neumann boundary conditions in terms of the
spectral $\zeta$-function of the piston. As a particular case, we specify the
piston to be a $d$-dimensional sphere and present explicit results for
$d=2,3,4,5$.
