Dirac and Weyl semimetals, materials where electrons behave as relativistic fermions, react to position- and time-dependent perturbations, such as strain, as if emergent electromagnetic fields were applied. Since they differ from external electromagnetic fields in their symmetries and phenomenology they are called pseudo-electromagnetic fields, and enable a simple and unified description of a variety of inhomogeneous systems involving topological semimetals. We review the different physical ways to create effective pseudo-fields, their observable consequences as well as their similarities and differences compared to electromagnetic fields. Among these difference is their effect on quantum anomalies, the absence of a classical symmetry in the quantum theory, which we revisit from a quantum field theory and a semiclassical viewpoint. We conclude with predicted observable signatures of the pseudo-fields and the nascent experimental status.