Divestment from companies that produce or heavily utilise fossil fuels has become one of the biggest issues in the contemporary moral debate surrounding climate change. Pressured by student and civil society based movements, universities and other institutions are being asked to divest themselves of their fossil fuel related investments. To date, there have been some very major institutions and funds who have begun to divest, the most notable being Norway’s Government Pension Fund. Many universities have also followed suit. There are many issues at play in the divestment debate, but one fundamental issue concerns the moral arguments that are used to justify divestment. Given that divestment is one of the biggest issues and fastest growing movements in the climate debate, it is important to be clear on these arguments. In what follows, I will argue that the case for divestment is morally strong. Nonetheless, there is a plethora of arguments often used to support divestment, not all of which have equal strength. Moreover, it matters a great deal for the strength of the conclusions regarding divestment which of those arguments are employed. There are two major groups of arguments: what I call positive arguments for divestment, which stem from the claim that divestment is a response to general duties to take action to prevent dangerous climate change and, negative arguments, which stem from a duty not to cause harm. There are also ‘internal’ and economic arguments. In this paper I will briefly characterise what divestment means and to whom it applies. I will then look at negative and positive duty accounts of the duty to divest before looking at the standard objections to both.