In this talk, I will give a survey of results from the last few years, all having to do with maximal discrete sets; such are, for example, MAD families, maximal cofinitary groups, or bases in vector spaces. The underlying theme is, I claim, that these objects allow us to study the relationships between definability, regularity (that is, generalised notions of measure), large cardinals, and forcing axioms; namely, by their ``higher-dimensional'' nature, they can make visible many new relationships between these classical themes.