Topological physics has been extensively studied in electrical circuits, where the resemblance between circuit Laplacians and tight-binding models allows for the exploration of exotic topological phases. This review discusses the circuit realization of topological insulators (TIs) and semimetals (TSMs), as well as unconventional topological states such as non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional states. The key idea is to map tight-binding Hamiltonians to circuit Laplacians, enabling the realization of a wide variety of topological states that are challenging to observe in conventional condensed matter systems. The flexibility of electrical circuits allows for the study of these states, including their interactions and behaviors in higher-dimensional, non-periodic, and non-Euclidean lattices. Additionally, circuits can be used to simulate physical phenomena in other systems, such as photonic and magnetic ones. The compatibility of TECs with traditional integrated circuits makes them convenient for manufacture and miniaturization. This review outlines the fundamentals of TECs, including circuit elements, equations, and construction methods, and discusses the circuit realization of various topological states. It also highlights the potential applications of TECs in exploring topological physics, (meta)material designs, and device applications.Topological physics has been extensively studied in electrical circuits, where the resemblance between circuit Laplacians and tight-binding models allows for the exploration of exotic topological phases. This review discusses the circuit realization of topological insulators (TIs) and semimetals (TSMs), as well as unconventional topological states such as non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional states. The key idea is to map tight-binding Hamiltonians to circuit Laplacians, enabling the realization of a wide variety of topological states that are challenging to observe in conventional condensed matter systems. The flexibility of electrical circuits allows for the study of these states, including their interactions and behaviors in higher-dimensional, non-periodic, and non-Euclidean lattices. Additionally, circuits can be used to simulate physical phenomena in other systems, such as photonic and magnetic ones. The compatibility of TECs with traditional integrated circuits makes them convenient for manufacture and miniaturization. This review outlines the fundamentals of TECs, including circuit elements, equations, and construction methods, and discusses the circuit realization of various topological states. It also highlights the potential applications of TECs in exploring topological physics, (meta)material designs, and device applications.