Justice systems are sometimes called upon to evaluate cases in which healthcare professionals are suspected of killing their patients illegally. These cases are difficult to evaluate because they involve at least two levels of uncertainty. Commonly in a murder case it is clear that a homicide has occurred, and investigators must resolve uncertainty about who is responsible. In the cases we examine here there is also uncertainty about whether homicide has occurred. Investigators need to consider whether the deaths that prompted the investigation could plausibly have occurred for reasons other than homicide, in addition to considering whether, if homicide was indeed the cause, the person under suspicion is responsible. In this report, the RSS provides advice and guidance on the investigation and evaluation of such cases. This report was prompted by concerns about the statistical challenges such cases pose for the legal system. The cases often turn, in part, on statistical evidence that is difficult for lay people and even legal professionals to evaluate. Furthermore, the statistical evidence may be distorted by biases, hidden or apparent, in the investigative process that render it misleading. In providing advice on how to conduct investigations in such cases, this report particularly focuses on minimising the kinds of biases that could distort statistical evidence arising from the investigation. This report also provides guidance on how to recognise and take account of such biases when evaluating statistical evidence and more broadly on how to understand the strengths and limitations of such evidence and give it proper weight. This report is designed specifically to help all professionals involved in investigating such cases and those who evaluate such cases in the legal system, including expert witnesses. It will also be of interest to scholars and legal professionals who are interested in the role of statistics in evidentiary proof, and more generally to anyone interested in improving criminal investigations. With such a wide range of audiences, it is inevitable that for some readers certain sections may seem more relevant, and some less so, but we believe it is important not to aim particular sections at particular kinds of reader. We want, for example, the barrister to see what advice we give to the expert statistical witness – and we hope understand it, at least in broad terms – and vice versa; we believe that is important in helping all parties to appreciate the contributions of others in reaching just outcomes.