document: Complex Langevin (CL) dynamics [1,2] provides an approach to circumvent the sign problem in numerical simulations of lattice field theories with a complex Boltzmann weight, since it does not rely on importance sampling. In recent years a number of stimulating results has been obtained in the context of nonzero chemical potential, in both lower and four-dimensional field theories with a severe sign problem in the thermodynamic limit [3–8] (for two recent reviews, see e.g. Refs. [9,10]). However, as has been known since shortly after its inception, correct results are not guaranteed [11–16]. This calls for an improved understanding, relying on the combination of analytical and numerical insight. In the recent past, the important role played by the properties of the real and positive probability distribution in the complexified configuration space, which is effectively sampled during the Langevin process, has been clarified [17,18]. An important conclusion was that this distribution should be sufficiently localised in order for CL to yield valid results. Importantly, this insight has recently also led to promising results in nonabelian gauge theories, with the implementation of SL(N,C) gauge cooling [8,10].