Estimation fusion, or data fusion for estimation, is the problem of how to best utilize useful information contained in multiple data sets for the purpose of estimating an unknown quantity — a parameter or a process. Estimation fusion with constraints gives rise to challenging theoretical problems given the observations from multiple geometrically dispersed sensors: Under dimensionality constraints, how to preprocess data at each local sensor to achieve the best estimation accuracy at the fusion center? Under communication bandwidth constraints, how to quantize local sensor data to minimize the estimation error at the fusion center? Under constraints on storage, how to optimally update state estimates at the fusion center with out-of-sequence measurements? Under constraints on storage, how to apply the out-of-sequence measurements (OOSM) update algorithm to multi-sensor multi-target tracking in clutter? The present work is devoted to the above topics by applying the best linear unbiased estimation (BLUE) fusion. We propose optimal data compression by reducing sensor data from a higher dimension to a lower dimension with minimal or no performance loss at the fusion center. For single-sensor and some particular multiple-sensor systems, we obtain the explicit optimal compression rule. For a multisensor system with a general dimensionality requirement, we propose the Gauss-Seidel iterative algorithm to search for the optimal compression rule. Another way to accomplish sensor data compression is to find an optimal sensor quantizer. Using BLUE fusion rules, we develop optimal sensor data quantization schemes according to the bit rate constraints in communication between each sensor and the fusion center. For a dynamic system, how to perform the state estimation and sensor quantization update simultaneously is also established, along with a closed form of a recursion for a linear system with additive white Gaussian noise. A globally optimal OOSM update algorithm and a constrained optimal update algorithm are derived to solve one-lag as well as multi-lag OOSM update problems. In order to extend the OOSM update algorithms to multisensor multitarget tracking in clutter, we also study the performance of OOSM update associated with the Probabilistic Data Association (PDA) algorithm.