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Vital technology trends such as voltage scaling and homogeneous multicore scaling have reached their limits and architects turn to alternate computing paradigms, such as heterogeneous and domain-specialized solutions. Coarse-Grain Reconfigurable Arrays (CGRAs) promise the performance of massively spatial computing while offering interesting trade-offs of flexibility versus energy efficiency. Yet, configuring and scheduling execution for CGRAs generally runs into the classic difficulties that have hampered Very-Long Instruction Word (VLIW) architectures: efficient schedules are difficult to generate, especially for applications with complex control flow and data structures, and they are inherently static - thus, in adapted to variable-latency components (such as the read ports of caches). Over the years, VLIWs have been relegated to important but specific application domains where such issues are more under the control of the designers; similarly, statically-scheduled CGRAs may prove inadequate for future general-purpose computing systems. In this paper, we introduce Elastic CGRAs, the superscalar processors of computing fabrics: no complex schedule needs to be computed at configuration time, and the operations execute dynamically in the CGRA when data are ready, thus exploiting the data parallelism that an application offers. We designed, down to a manufacturable layout, a simple CGRA where we demonstrated and optimized our elastic control circuitry. We also built a complete compilation toolchain that transforms arbitrary C code in a configuration for the array. The area overhead (26.2%), critical path overhead (8.2%) and energy overhead (53.6%) of Elastic CGRAs over non-elastic CGRAs are significantly lower than the overhead of superscalar processors over VLIWs, while providing the same benefits. At such moderate costs, elasticity may prove to be one of the key enablers to make the adoption of CGRAs widespread.