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In this paper we present a novel formalism for semantic construction called DUDES (Dependency-based Underspecified Discourse REpresentation Structures). The DUDES formalism has been designed to overcome the rigidity of semantic composition based on the lambda calculus (where the order of application is typically fixed) and provides some flexibility with respect to the direction of the dependence and with respect to the order of application of arguments. In this short paper we present the DUDES formalism and work through a simple example. DUDES bears some resemblance to the work on λ-DRT [2] and LUDs [1] as well as with the work of Copestake et al. [4] and represents a generalization of the formalism introduced in [3]. A detailed discussion of the relation to these formalisms is clearly out of the scope of this paper. DUDES are characterized by three main facts. First, they represent semantic dependencies and are thus inherently suitable for a dependency-based grammar formalism assuming that syntactic dependencies correspond to semantic dependencies (though the correspondence might be "inverted"). Second, they explicitly encode scope relations and are thus able to yield underspecified representations as output (in contrast to the linear logic approach for LFG [5] where different scopings correspond to different derivations). Third, there is one single operation for semantic composition which is, to some extent, order-independent (in contrast to traditional lambda-based formalisms) as well as flexible with respect to the direction of the syntactic dependency. As the name suggests, DUDES builds on DRT [6] and in particular on UDRT [7] in the sense that it relies on labeled DRSs and dominance relations between these to talk about scope.