This overview paper defends an augmented cognitively oriented generic-design hypothesis: there are both significant similarities between the design activities implemented in different situations and crucial differences between these and other cognitive activities; yet, characteristics of a design situation (related to the design process, the designers, and the artefact) introduce specificities in the corresponding cognitive activities and structures that are used, and in the resulting designs. We thus augment the classical generic-design hypothesis with that of different forms of designing. We review the data available in the cognitive design research literature and propose a series of candidates underlying such forms of design, outlining a number of directions requiring further elaboration.