The nature of abstract and concrete semantics and differences between them have remained a debated issue in psycholinguistic and cognitive studies for decades. Most of the available behavioral and neuroimaging studies reveal distinctions between these two types of semantics, typically associated with a so-called “concreteness effect.” Many attempts have been made to explain these differences using various approaches, from purely theoretical linguistic and cognitive frameworks to neuroimaging experiments. In this brief overview, we will try to provide a snapshot of these diverse views and relationships between them and highlight the crucial issues preventing this problem from being solved. We will argue that one potentially beneficial way forward is to identify the neural mechanisms underpinning acquisition of the different types of semantics (e.g., by using neurostimulation techniques to establish causal relationships), which may help explain the distinctions found between the processing of concrete and abstract semantics.