Most neuroscientists would agree that for brain research to progress, we have to know which experimental manipulations have no effect as much as we must identify those that do have an effect. The dominant statistical approaches used in neuroscience rely on P values and can establish the latter but not the former. This makes non-significant findings difficult to interpret: do they support the null hypothesis or are they simply not informative? Here we show how Bayesian hypothesis testing can be used in neuroscience studies to establish both whether there is evidence of absence and whether there is absence of evidence. Through simple tutorial-style examples of Bayesian t-tests and ANOVA using the open-source project JASP, this article aims to empower neuroscientists to use this approach to provide compelling and rigorous evidence for the absence of an effect.