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Abstract: During the 5th Framework Program (FP5) the European Commission (EC) funded the PDS-XADS project (Preliminary Design Studies on an eXperimental Accelerator Driven System) mainly devoted to the demonstration of the feasibility of the coupling among an accelerator, a spallation target and a subcritical core, and to preliminarily investigate the conditions for enhancing the transmutation rate at values suitable for an industrial-scale transmuter. The on-going activities of the FP6 EUROTRANS integrated project (EUROpean research programme for TRANSmutation of high level nuclear waste in an accelerator driven system) are in continuity with the PDS-XADS project objectives, but with a significant change for what concerns the final goal in terms of demonstration level. In fact they are oriented to both the design of an industrial-scale transmuter (EFIT) and a small experiment facility (XT-ADS), serving as a test bench for as much component technologies as possible in a realistic irradiation environment. The paper describes the main results of the design activities for the large and small LBE-cooled cores performed within PDS-XADS project and discusses the constraints (e.g. those deriving from the spallation target and the accelerator), the requirements (such as the degree of subcriticality when operating at power as well as during refuelling), the design criteria and choices with the purpose to verify their applicability to new designs (EFIT, XT-ADS). [Copyright &y& Elsevier]