Lexical semantic competence is a multifaceted and complex reality, which includes the ability of drawing inferences, distinguishing different word senses, referring to the entities in the world, and so on. A long-standing tradition of research in linguistics and cognitive science has investigated these issues using symbolic representations. The aim of this chapter is to understand how and to what extent the major aspects of lexical meaning can be addressed with distributional representations. We have selected a group of research topics that have received particular attention in distributional semantics: (i) identifying and representing multiple meanings of lexical items, (ii) discriminating between different paradigmatic semantic relations, (iii) establishing cross-lingual links among lexemes, (iv) analyzing connotative aspects of meaning, (v) studying semantic change, (vi) grounding distributional representations in extralinguistic experiential data, and (vii) using distributional vectors in cognitive science to model the mental lexicon and semantic memory.