Artificial neurons and synapses are considered essential for the progress of the future brain‐inspired computing, based on beyond von Neumann architectures. Here, a discussion on the common electrochemical fundamentals of biological and artificial cells is provided, focusing on their similarities with the redox‐based memristive devices. The driving forces behind the functionalities and the ways to control them by an electrochemical‐materials approach are presented. Factors such as the chemical symmetry of the electrodes, doping of the solid electrolyte, concentration gradients, and excess surface energy are discussed as essential to understand, predict, and design artificial neurons and synapses. A variety of two‐ and three‐terminal memristive devices and memristive architectures are presented and their application for solving various problems is shown. The work provides an overview of the current understandings on the complex processes of neural signal generation and transmission in both biological and artificial cells and presents the state‐of‐the‐art applications, including signal transmission between biological and artificial cells. This example is showcasing the possibility for creating bioelectronic interfaces and integrating artificial circuits in biological systems. Prospectives and challenges of the modern technology toward low‐power, high‐information‐density circuits are highlighted.