In 1972, Efim Liberman, a Soviet biophysicist, pioneered a brand-new approach to studying the operation of the brain, the live cell and the human mind by publishing a paper titled “Cell as a molecular computer” (1972). In this paper, Liberman posited that a consecutive/parallel stochastic molecular computer (MCC) controls a living cell. An MCC operates with molecule-words (DNA, RNA, proteins) according to the program recorded in DNA and RNA. Computational operations are implemented by molecular operators acting as enzymes. An MCC is present in each live cell. A neuron cell MCC can be involved in solving tasks for the entire organism. Neuron MCC investigation was started with studying an impact of an intracellular injection of cyclic AMP on electric activity of a neuron. Cyclic nucleotides were considered as input words for an MCC, which are generated inside a neuron as a result of synaptic activity. This led Efim Liberman to the idea that, in order to solve complex physical problems, which are encountered by a neuron and require rapid solutions, the molecular computer adjusts the operation of the quantum molecular regulator, which uses the “computational environment” of the cytoskeleton and quantum properties of the elementary hypersound quasiparticles for completing mathematical operations for the minimum price of action. Efim Liberman suggested that the human self-consciousness is a quantum computer of even a higher level and designated it as an extreme quantum regulator. In order to describe such systems, he suggested to join biology, physics and mathematics into a unified science, and formulated its four fundamental principles. Results of Efim Liberman’s theoretical and experimental studies on the topic of biological computation are summarized in this review.