Abstract: We will consider measurable versions of classical combinatorial problems (vertex/edge colourings, matchings, etc) and their applications. The main object of study will be bounded-degree graphings (that is, graphs whose vertex set is a standard probability space and whose edge set is the union of finitely many measure-preserving matchings). Graphings appear in various areas such as the limit theory of bounded-degree graphs, measure-preserving group actions, descriptive set theory, etc. The existence of a measurable function F that satisfies given combinatorial constraints (such as being a proper vertex colouring) is of interest because it may be used, for example, to distinguish non-isomorphic graphings or be transferred to finite graphs in the context of property testing. We will mostly concentrate on positive results. Here, a powerful tool for constructing the desired function F is to design a parallel decentralised algorithm that converges to it almost everywhere.