  The inclusion of non-Abelian U(N) internal charges (other than the electric
charge) into Twistor Theory is accomplished through the concept of "colored
twistors" (ctwistors for short) transforming under the colored conformal
symmetry U(2N,2N). In particular, we are interested in 2N-ctwistors describing
colored spinless conformal massive particles with phase space
U(2N,2N)/[U(2N)xU(2N)]. Penrose formulas for incidence relations are
generalized to N>1. We propose U(2N)-gauge invariant Lagrangians for
2N-ctwistors and we quantize them through a bosonic representation,
interpreting quantum states as particle-hole excitations above the ground
state. The connection between the corresponding Hilbert (Fock-like with
constraints) space and the carrier space of a discrete series representation of
U(2N,2N) is established through a coherent space (holomorphic) representation.
