In this thesis we study new anisotropic variational regularisers and partial differential equations (PDEs) for solving inverse imaging problems that arise in a variety of real-world applications. Firstly, we introduce a new anisotropic higher-order total directional variation regulariser. We describe both the theoretical and the numerical details for its use within a variational formulation for solving inverse problems and give examples for the reconstruction of noisy images and videos, image zooming and the interpolation of scattered surface data. Secondly, we focus on a non-symmetric drift-diffusion equation, called osmosis. We propose an efficient numerical implementation of the osmosis equation, based on alternate directions and operator splitting techniques. We study their scale-space properties and show their efficiency in processing large images. Moreover, we generalise the osmosis equation to accommodate suitable directional information: this modification turns out to be useful to correct for the well-known blurring artefacts the original osmosis model introduces when applied to shadow removal in images. Last but not least, we explore applications of variational models and PDEs to cultural heritage conservation. We develop a new non-invasive technique that uses multi-modal imaging for detecting sub-superficial defects in fresco walls at sub-millimetre precision. We correct light-inhomogeneities in these imaging measurements that are due to measurement errors via osmosis filtering, in particular making use of the efficient computational schemes that we introduced before for dealing with the large-scale nature of these measurements. Finally, we propose a semi-supervised workflow for the detection and inpainting of defects in damaged illuminated manuscripts.