D-LTAG is a discourse-level extension of lexicalized tree-adjoining grammar (LTAG), in which discourse syntax is projected by different types of discourse connectives and discourse interpretation is a product of compositional rules, anaphora resolution, and inference. In this paper, we present a D-LTAG extension of ongoing work on an LTAG syntax-semantic interface. First, we show how predicate-argument semantics are computed for standard, ‘structural’ discourse connectives. These are connectives that retrieve their semantic arguments from their D-LTAG syntactic tree. Then we focus on discourse connectives that occur syntactically as (usually) fronted adverbials. These connectives do not retrieve both their semantic arguments from a single D-LTAG syntactic tree. Rather, their predicate-argument structure and interpretation distinguish them from structural connectives as well as from other adverbials that do not function as discourse connectives. The unique contribution of this paper lies in showing how compositional rules and anaphora resolution interact within the D-LTAG syntax-semantic interface to yield their semantic interpretations, with multi-component syntactic trees sometimes being required.