We introduce a new class of higher-order total directional variation regularizers. These regularizers extend the directional total variation for a constant direction \cite{directionaltv} to accommodate spatially-adaptive directions and higher-order derivatives. For the numerical solution of the resulting variational regularisation approach, we propose a primal-dual hybrid gradient approach whose details we report in the paper. We demonstrate that the proposed regulariser is able to preserve and enhance intrinsic anisotropic features in images by showing results for different imaging applications: image denoising, wavelet-based image zooming and reconstruction of surfaces from scattered height measurements.