Excited $D$ and $D_s$ meson spectroscopy from lattice QCD

We present highly excited spectra of charm-light and charm-strange mesons from dynamical lattice QCD. Our calculations are performed on anisotropic $N_f=2+1$ dynamical ensembles generated by the Hadron Spectrum Collaboration. The use of novel techniques and a large basis of interpolating operators have allowed us to extract these spectra to a high degree of statistical precision, extract states of high spin and observe candidate hybrid mesons. We interpret and discuss our results in light of the current experimental situation.