Dirac quantum materials comprise a broad category of condensed matter systems characterized by low-energy excitations described by the Dirac equation. These excitations, which can manifest as either collective states or band structure effects, have been identified in a wide range of systems, from exotic quantum fluids to crystalline materials. Over the past several decades, they have sparked extensive experimental and theoretical investigations in various materials, such as topological insulators and topological semimetals. The study of Dirac quantum materials has also opened up new possibilities for topological quantum computing, giving rise to a burgeoning field of physics and offering a novel platform for realizing rich topological phases, including various quantum Hall effects and topological superconducting phases. Furthermore, the topologically non-trivial band structures of Dirac quantum materials give rise to plentiful intriguing transport phenomena, including longitudinal negative magnetoresistance, quantum interference effects, helical magnetic effects, and others. Currently, numerous transport phenomena in Dirac quantum materials remain poorly understood from a theoretical standpoint, such as linear magnetoresistance in weak fields, anomalous Hall effects in nonmagnetic materials, and three-dimensional quantum Hall effects. Studying these transport properties will not only deepen our understanding of Dirac quantum materials, but also provide important insights for their potential applications in spintronics and quantum computing. In this paper, quantum transport theory and quantum anomaly effects related to the Dirac equation are summarized, with emphasis on massive Dirac fermions and quantum anomalous semimetals. Additionally, the realization of parity anomaly and half-quantized quantum Hall effects in semi-magnetic topological insulators are also put forward. Finally, the key scientific issues of interest in the field of quantum transport theory are reviewed and discussed.