All minerals behave elastically; elasticity is a rheological property that controls their ability to support stress, strain, and pressure; controls the nature of acoustic wave propagation; and influences subsequent plastic (i.e. permanent non-reversible) deformation. All minerals are intrinsically anisotropic in their elastic properties – that is, they have directional variations that are related to the configuration of the crystal lattice. This means that the commonly used mechanical elastic properties that relate elastic stress to elastic strain, including Young's modulus (E), Poisson's ratio (ν), shear modulus (G) and linear compressibility (β), are dependent on crystallographic direction. In this paper, we explore the ranges of anisotropy of E, ν, G and β in 86 rock-forming minerals, using previously published data, and show that the range is much wider than commonly assumed. We also explore how these variations (the directionality and the magnitude) are important for fundamental processes in the solid earth, including deformation (mechanical) twinning, coherent phase transformations and brittle failure. We present a new open-source software package (AnisoVis, written in MATLAB), which we use to calculate and visualise directional variations in elastic properties of rock-forming minerals. Following previous work in the fields of chemistry and materials science, we demonstrate that by visualising the variations in elasticity, we discover previously unreported properties of rock-forming minerals. For example, we show previously unreported directions of negative Poisson's ratio and negative linear compressibility, and we show that the existence of these features is more widespread (i.e. present in many more minerals) than previously thought. We illustrate the consequences of intrinsic elastic anisotropy for the elastic normal and shear strains within α-quartz single crystal under different applied stress fields; the role of elastic anisotropy on Dauphiné twinning and the α–β phase transformations in quartz; and stress distributions around voids of different shapes in talc, lizardite, albite, and sanidine. In addition to our specific examples, elastic anisotropy in rock-forming minerals, to the degree that we describe, has significant consequences for seismic (acoustic) anisotropy, for the focal mechanisms of earthquakes in anisotropic source regions (e.g. subducting slabs), for a range of brittle and ductile deformation mechanisms in minerals, and for geobarometry using mineral inclusions.