Quantum fluctuations in an interacting, classically-conformal field theory often break the underlying conformal invariance. This phenomenon, known as the conformal (as well as "scale," "Weyl," or "trace") anomaly, reveals itself via the presence of a nonzero beta function which determines the dependence of a particular coupling constant on the energy scale of the interaction. In this talk, we review the consequences of the conformal anomaly in solid state systems, focusing on a Dirac semimetal as an example. We show that the conformal anomaly, associated with the electric charge renormalization in the material, produces a thermal Nernst current, leads to a polynomial screening of an external electrostatic field, and interferes naturally with the axial anomaly. Relations to field-theoretical phenomena, such as the Schwinger pair production and transport phenomena in gravitational/electromagnetic field backgrounds, are highlighted.