The generation of polarization under mechanical stress or, conversely, the generation of deformation in response to a voltage, are a traditional preserve of ferroelectric and piezoelectric materials. Piezoelectricity, however, is not the only electromechanical property of materials. Flexoelectricity, the coupling between polarization and strain gradients, is increasingly being recognized as a very important, sometimes even dominant, electromechanical property. Contrary to piezoelectricity, flexoelectricity is symmetry-allowed in all materials, including of course ferroelectrics, but also centrosymmetric materials. It turns out also that this property even exists in semiconductors and photovoltaic materials, metals, and biomaterials. It is thus truly universal. Moreover, because strain gradients tend to grow in inverse proportion to physical size, flexoelectricity can be the dominant electromechanical coupling at the nanoscale. Additionally, when it coexists with other functional properties (piezoelectricity in ferroelectrics, photovoltaic effects in semiconductors), it generates new functional properties that would not be possible without such coexistence. Understanding flexoelectricity is therefore important and potentially useful if we manage to harness these new flexoelectrically-induced functionalities. The present tutorial will attempt to give a complete overview of flexoelectricity, from its definition and physical principles, to its effect on different materials and the consequences for devices. At every step of the way, the working principles will be illustrated with examples. The aim is to provide a good foundation for those who are new or curious about flexoelectricity, and perhaps also to inspire new directions of research for those who already primed about it.