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We present diffusion algorithms for distributed estimation and detection over networks that endow all nodes with both spatial cooperation abilities and temporal processing abilities. Each node in the network is allowed to share information locally with its neighbors; this step amounts to sharing and processing of spatial data. At the same time, each node is allowed to after and process past estimates to improve estimation accuracy through an overall collaborative process. In this manner, the resulting distributed algorithms consist of three stages: adaptation, spatial processing, and temporal processing. Moreover, the order of these three stages can be interchanged leading to a total of six variations. The results indicate that whether temporal processing is performed before or after adaptation, the strategy that performs adaptation before spatial cooperation leads to smaller mean-square error. The additional temporal processing step is useful in combating perturbations due to noise over the communications links. We further describe an application in the context of distributed detection and provide computer simulations to illustrate and support the findings.