This thesis presents a unique and efficient solver to the state estimation (SE) problem for the power grid, based on probabilistic graphical models (PGMs). SE is a method of estimating the varying state values of voltage magnitude and phase at every bus within a power grid based on meter measurements. However, existing SE solvers are notorious for their computational inefficiency to calculate the matrix inverse, and hence slow convergence to produce the final state estimates. The proposed PGM-based solver estimates the state values from a different perspective. Instead of calculating the matrix inverse directly, it models the power grid as a PGM, and then assigns potentials to nodes and edges of the PGM, based on the physical constraints of the power grid. This way, the original SE problem is transformed into an equivalent probabilistic inference problem on the PGM, for which two efficient algorithms are proposed based on Gaussian belief propagation (GBP). The equivalence between the proposed PGM-based solver and existing SE solvers is shown in terms of state estimates, and it is experimentally demonstrated that this new method converges much faster than existing solvers.