This paper is concerned with the power system state estimation (PSSE) problem that aims to find the unknown operating point of a power network based on a given set of measurements. We first study the power flow (PF) problem as an important special case of PSSE. PF is known to be nonconvex and NP-hard in the worst case. To this end, we propose a set of semidefinite programs (SDPs) with the property that they all solve the PF problem as long as the voltage angles are relatively small. Associated with each SDP, we explicitly characterize the set of all the complex voltages that can be recovered via that convex problem. As a generalization, the design of an SDP problem that recovers multiple nominal points and a neighborhood around each point is also cast as a convex program. The results are, then, extended to the PSSE problem, where the measurements used in the PF problem are subject to noise. A two-term objective function is employed for each convex program developed for the PSSE problem: 1) the first term accounting for the nonconvexity of the PF equations and 2) other one for estimating the noise levels. An upper bound on the estimation error is derived with respect to the noise level, and the proposed techniques are demonstrated on multiple test systems, including a 9241-bus European network. Although the focus of this paper is on power networks, yet the developed results apply to every arbitrary state estimation problem with quadratic measurement equations.