An important goal of cognitive neurosciences is to understand the functional organization of the brain. It heavily relies on Functional Magnetic Resonance Imaging (fMRI), a powerful tool to investigate the link between brain function and anatomical structures at a high spatial-resolution. Functional inter-individual variability is a major obstacle limiting functional brain mapping precision and generalizability of results obtained in neuroimaging studies. This variability, observed across subjects performing the same task, goes far beyond anatomical variability in brain shape and size. In this work, we focus on a class of methods designed to address functional variability, namely functional alignment. These methods match subjects neural signals based on their functional similarity.In a first part, we review standard functional brain mapping paradigms and techniques, as well as the challenges induced by functional variability. We additionally review existing functional alignment methods and related work, and discuss the current limitations of these approaches. In a second part, we develop a new functional alignment method, based on optimal transport - a mathematical theory interested in matching probability distributions while taking their geometry into account. Functional alignment methods are local, which means that many local alignments need to be aggregated to compose whole-brain alignments. Moreover, these methods derive pairwise matching and call for a “functional template”, a common functional representation to which all subjects of a study can be aligned. To overcome limitations of existing solutions, we additionally introduce a new aggregation scheme as well as a principled template design procedure. In a third part, we turn to empirical validation of alignment performance. Indeed, these methods are seldom used in applied studies, and it is unclear to what extent they can address functional variability in typical cognitive studies. We investigate their performance to improve generalization of predictive models to new subjects. In this inter-subject decoding set-up, spanning four different datasets, we show that alignment methods hold real potential to recover an important share of prediction accuracy lost due to inter-subject variability.