The article provides a review of foreign scientific literature on existing approaches in scientists' ideas about the quantum theory of brain physics and the theory of quantum-like consciousness. It is shown that in recent years, physics and psychology have made important scientific advances; the emergence of new tools has led to the development of neuroscience, which makes it possible to study the state of the brain when known mental functions are preserved or impaired. The use of mathematical methods has made it possible to develop computational models that describe the fundamental functions of the brain and implement them in technological applications. The most pressing issues of modern research in the field of mental processes of the brain, such as perception, cognition, emotions, consciousness and learning, researchers call the problems of the universal properties of human behavior in various conditions and cultures; the potential of computational psychiatry to predict, understand and treat mental disorders; neural mechanisms of influence on mental content at higher levels of organization of the hierarchical brain; mechanisms of the emergence of abstract ideas from sensorimotor accidents, etc. Currently, in understanding the cognitive processes of the brain, perception, understanding, cognition, mental lexicon, understanding the generation and nature of emotions, memory, learning, etc. A number of researchers use the theory of quantum or quantum-like consciousness. Research in this area is both empirical and theoretical in nature, with the postulates of the theory being considered mainly related to quantum physics due to the lack of other areas in which quantum-like properties are supposedly manifested. It is believed that despite modern advances in artificial intelligence models, neural networks cannot reach the level of the human brain, partly due to differences in the way information is encoded and processed. In order to develop the potential of brain-based artificial intelligence, quantum neural networks are proposed that are similar to the mechanisms and phenomena in the brain. Interactions between biologically important molecules in the brain may underlie the quantum metalanguage that governs the behavior of humans and animals.