  We give some general results on proper-biharmonic submanifolds of a complex
space form and, in particular, of the complex projective space. These results
are mainly concerned with submanifolds with constant mean curvature or parallel
mean curvature vector field. We find the relation between the bitension field
of the inclusion of a submanifold $\bar{M}$ in $\mathbb{C}P^n$ and the
bitension field of the inclusion of the corresponding Hopf-tube in
$\mathbb{S}^{2n+1}$. Using this relation we produce new families of
proper-biharmonic submanifolds of $\mathbb{C}P^n$. We study the geometry of
biharmonic curves of $\mathbb{C}P^n$ and we characterize the proper-biharmonic
curves in terms of their curvatures and complex torsions.
