Response inhibition is the ability to override a planned or an already initiated response. It is the hallmark of executive control as its deficits favour impulsive behaviours, which may be detrimental to an individual's life. This article reviews behavioural and computational guises of response inhibition. It focuses only on inhibition of oculomotor responses. It first reviews behavioural paradigms of response inhibition in eye movement research, namely the countermanding and antisaccade paradigms, both proven to be useful tools for the study of response inhibition in cognitive neuroscience and psychopathology. Then, it briefly reviews the neural mechanisms of response inhibition in these two behavioural paradigms. Computational models that embody a hypothesis and/or a theory of mechanisms underlying performance in both behavioural paradigms as well as provide a critical analysis of strengths and weaknesses of these models are discussed. All models assume the race of decision processes. The decision process in each paradigm that wins the race depends on different mechanisms. It has been shown that response latency is a stochastic process and has been proven to be an important measure of the cognitive control processes involved in response stopping in healthy and patient groups. Then, the inhibitory deficits in different brain diseases are reviewed, including schizophrenia and obsessive-compulsive disorder. Finally, new directions are suggested to improve the performance of models of response inhibition by drawing inspiration from successes of models in other domains. This article is part of the themed issue `Movement suppression: brain mechanisms for stopping and stillness'.