State estimation is a fundamental task in power system monitoring. The focus in this thesis is on Distribution System State Estimation (DSSE). One of the main challenges in DSSE is the lack of observability due to the small number of sensor installations in practical power distribution circuits, where the number of measurements is far fewer than the number of state variables. In this thesis, our goal is to develop DSSE methods which address the low-observability challenges.First, we leverage the high reporting rate of a small number of distribution-level phasor measurement units (D-PMUs), a.k.a., micro-PMUs, to unmask and characterize sparsity patterns among the state variables in radial power distribution systems. Accordingly, the DSSE problem is formulated over the differential synchrophasors as an adaptive group sparse recovery problem to track the changes that are made in the states of the system and captured by D-PMU measurements. To enhance the performance of the proposed method, the formulated DSSE is further augmented by the side information on the support of the vector of unknowns that is obtained from the outcome of an event-zone identification analysis prior to solving the DSSE problem. Second, to capture the dynamics of the power distribution system, we model the DSSE problem under an event-triggered setting, where we use the estimations of the state variables during the previous events as priori information to predict the state variables at the current event. Accordingly, a novel data-driven method based on elastic net regression is proposed to learn the event-triggered state transition matrix; despite the low-observability in the system. Here, in the absence of direct power measurements, we enhance our ability in sparse recovery by developing a new reinforced physics-based coupling method among the state variables, in which we add a novel set of linear differential power flow equations to the DSSE problem formulation in forms of virtual measurements. Third, we study the joint estimation of sensitivity distribution factors and power flows in low-observable power distribution systems by developing a novel physics-aware measurement-based approach that takes into account the sparsity features of the problem extracted for radial power distribution systems.