Summary: The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.