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Ontologies and vector space embeddings are among the most popular frameworks for encoding conceptual knowledge. Ontologies excel at capturing the logical dependencies between concepts in a precise and clearly defined way. Vector space embeddings excel at modelling similarity and analogy. Given these complementary strengths, there is a clear need for frameworks that can combine the best of both worlds. In this paper, we present an overview of our recent work in this area. We first discuss the theory of conceptual spaces, which was proposed in the 1990s by Gärdenfors as an intermediate representation layer in between embeddings and symbolic knowledge bases. We particularly focus on a number of recent strategies for learning conceptual space representations from data. Next, building on the idea of conceptual spaces, we discuss approaches where relational knowledge is modelled in terms of geometric constraints. Such approaches aim at a tight integration of symbolic and geometric representations, which unfortunately comes with a number of limitations. For this reason, we finally also discuss methods in which similarity, and other forms of conceptual relatedness, are derived from vector space embeddings and subsequently used to support flexible forms of reasoning with ontologies, thus enabling a looser integration between embeddings and symbolic knowledge.