Recently, a new class of symmetry-protected topological insulators, higher-order topological insulator, has drawn research interest. Unlike conventional first-order topological insulators, two-dimensional second-order topological insulators have topologically protected corner states, and corresponding three-dimensional systems have topological gapless modes on the hinges. In some crystalline structures, the topological corner and hinge states can arise only from the nontrivial bulk topology when the lattice termination is compatible with the crystal symmetries. Recent investigations have shown that classical circuits exhibit great advantages in simulating these topological states. In this talk, we will introduce the research progress and some of our recent works for higher-order topological states based on circuits, including three-dimensional octupole insulators, four-dimensional hexadecapole insulators, higher-order topological Anderson insulators, non-Hermitian higher-order topological phases and their potential applications.