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Simultaneous localization and tracking (SLAT) in sensor networks aims to determine the positions of sensor nodes and a moving target in a network, given incomplete and inaccurate range measurements between the target and each of the sensors. One of the established methods for achieving this is to iteratively maximize a likelihood function (ML) of positions given the observed ranges, which requires initialization with an approximate solution to avoid convergence towards local extrema. This paper develops methods for handling both Gaussian and Laplacian noise, the latter modeling the presence of outliers in some practical ranging systems that adversely affect the performance of localization algorithms designed for Gaussian noise. A modified Euclidean distance matrix (EDM) completion problem is solved for a block of target range measurements to approximately set up initial sensor/target positions, and the likelihood function is then iteratively refined through majorization-minimization (MM). To avoid the computational burden of repeatedly solving increasingly large EDM problems in time-recursive operation, an incremental scheme is exploited whereby a new target/node position is estimated from previously available node/target locations to set up the iterative ML initial point for the full spatial configuration. The above methods are first derived under Gaussian noise assumptions, and modifications for Laplacian noise are then considered. Analytically, the main challenges to overcome in the Laplacian case stem from the non-differentiability of l1 norms that arise in the various cost functions. Simulation results show that the proposed algorithms significantly outperform existing localization methods in the presence of outliers, while providing comparable performance for Gaussian noise.