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Hardware/software (HW/SW) codesign and reconfigurable computing are commonly used methodologies for digital-systems design. However, no previous work has been carried out in order to define a HW/SW codesign methodology with dynamic scheduling for run-time reconfigurable architectures. In addition, all previous approaches to reconfigurable computing multicontext scheduling are based on static-scheduling techniques. In this paper, we present three main contributions: 1) a novel HW/SW codesign methodology with dynamic scheduling for discrete event systems using dynamically reconfigurable architectures; 2) a new dynamic approach to reconfigurable computing multicontext scheduling; and 3) a HW/SW partitioning algorithm for dynamically reconfigurable architectures. We have developed a whole codesign framework, where we have applied our methodology and algorithms to the case study of software acceleration. An exhaustive study has been carried out, and the obtained results demonstrate the benefits of our approach.