In this paper we describe how constraint modelling can support design innovation. Furthermore, we lay out how constraints are employed in the construction and exploration of a model's design space. We place the approach within the context of design exploration using computational and conceptual representations of design. A review of the literature reveals that geometric, topologic, functional, and quantitative constraints are those most commonly used. For each constraint type, an example is presented drawing from several workshops and research conducted by the author. The examples range from product design, to structural design, to fabrication issues in freeform geometry. Based on the case studies, we describe how the different types of constraints can be used as design drivers and help in the exploration of solution spaces. In conclusion, we identify the need for bidirectional exercising of constraints as the next challenge in design exploration and discuss how it is relevant in particular for cross domain design.