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 con- cerned 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 M ̄ in CPn and the bitension field of the inclusion of the corresponding Hopf-tube in S2n+1. Using this relation we produce new families of proper-biharmonic submanifolds of C P n . We study the geometry of biharmonic curves of C P n and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.