Anisotropy is a key structural feature of many physical processes. Despite this, most theory for the modelling and estimation of random fields is based on assuming isotropy of the observed field. Anisotropy can arise both in the structural features of the field, and between field components. I will discuss both forms of anisotropy, and how we may model them, parametrically for applications in geophysics such as understanding interface-loading processes, and more generically to capture strong directional preferences. I will also describe how we may nonparametrically identify the presence of anisotropic features without strong structural assumptions, such as a given parametric model class.