  Recently, topolectrical circuits (TECs) boom in studying the topological
states of matter. The resemblance between circuit Laplacians and tight-binding
models in condensed matter physics allows for the exploration of exotic
topological phases on the circuit platform. In this review, we begin by
presenting the basic equations for the circuit elements and units, along with
the fundamentals and experimental methods for TECs. Subsequently, we retrospect
the main literature in this field, encompassing the circuit realization of
(higher-order) topological insulators and semimetals. Due to the abundant
electrical elements and flexible connections, many unconventional topological
states like the non-Hermitian, nonlinear, non-Abelian, non-periodic,
non-Euclidean, and higher-dimensional topological states that are challenging
to observe in conventional condensed matter physics, have been observed in
circuits and summarized in this review. Furthermore, we show the capability of
electrical circuits for exploring the physical phenomena in other systems, such
as photonic and magnetic ones. Importantly, we highlight TEC systems are
convenient for manufacture and miniaturization because of their compatibility
with the traditional integrated circuits. Finally, we prospect the future
directions in this exciting field, and connect the emerging TECs with the
development of topology physics, (meta)material designs, and device
applications.
