This review article explores the recent advancements in topolectrical circuits (TECs), which have emerged as a promising platform for studying topological states of matter. TECs map tight-binding Hamiltonians from condensed matter physics to circuit Laplacians, allowing for the exploration of exotic topological phases. The authors begin by introducing the fundamental equations and methods for constructing TECs, including basic circuit elements and their equations. They then delve into the circuit realization of topological insulators (TIs) and semimetals (TSMs), highlighting the ability of TECs to observe unconventional topological states such as non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states. The review also discusses the use of TECs in exploring physical phenomena in other systems, such as photonic and magnetic circuits, and emphasizes the advantages of TECs in terms of manufacturing and miniaturization due to their compatibility with traditional integrated circuits. Finally, the authors outline future directions, connecting TECs to developments in topology physics, (meta)material designs, and device applications.This review article explores the recent advancements in topolectrical circuits (TECs), which have emerged as a promising platform for studying topological states of matter. TECs map tight-binding Hamiltonians from condensed matter physics to circuit Laplacians, allowing for the exploration of exotic topological phases. The authors begin by introducing the fundamental equations and methods for constructing TECs, including basic circuit elements and their equations. They then delve into the circuit realization of topological insulators (TIs) and semimetals (TSMs), highlighting the ability of TECs to observe unconventional topological states such as non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states. The review also discusses the use of TECs in exploring physical phenomena in other systems, such as photonic and magnetic circuits, and emphasizes the advantages of TECs in terms of manufacturing and miniaturization due to their compatibility with traditional integrated circuits. Finally, the authors outline future directions, connecting TECs to developments in topology physics, (meta)material designs, and device applications.