Despite the continuous growth of our knowledge of the functioning of nervous systems and sophisticated features of the neuroanatomy and neurophysiology of various animal species, the basic mechanisms that provide for such properties as the ability to learn, use memory, recognize patterns, and learn about the world are poorly understood. In this paper, we present an overview of artificial devices that model the brain and solve such cognitive tasks as navigation, pattern recognition, routing, and target site finding. We discuss both hybrid systems (hybrots), in which living neural networks control an artificial body, and systems in which such an artificial body is controlled by computer programs based on different models of the brain and its regions (animats). Two basic types of hybrid systems are considered: those in which the robot is connected to the brain of a living body, such as a rat, and those in which information is received from neurons taken from the body or neurons cultured on a microelectrode array detecting their electrical potentials. Among the computational approaches that simulate nervous systems of living organisms, we can mark out the Darwin family of devices based on the theory of neuronal group selection (TNGS). In addition, we consider papers in which animats solve navigation tasks using different models of the rat hippocampus, based on such modeling methods as cognitive graph, view cells, place cells, and experience cells. The approaches under consideration provide researchers with new tools to analyze basic principles of neuron interaction between each other and with the outside world, the principles that provide higher brain functions.