  We present spectra of highly excited D and Ds mesons up to around 3.8 GeV
determined using dynamical lattice QCD. We employ novel computational
techniques and the variational method with a large basis of carefully
constructed operators in order to extract and reliably identify the continuum
spin of an extensive set of excited states. These include states with high spin
and states identified as having an explicit gluonic contribution. Calculations
were performed on two volumes, both with a pion mass of approximately 400 MeV,
achieving a high statistical precision for both ground and excited states. We
discuss our results in light of experimental observations, comment on the
phenomenological implications and identify the lightest `supermultiplet' of
hybrid mesons in each sector.
