One of the most deeply rooted concepts in science is causality: the idea that events in the present are caused by events in the past and, in turn, act as causes for what happens in the future. If an event A is a cause of an effect B, then B cannot be a cause of A. Physicists, however, suspect that the notion of a definite causal order between events will be untenable in a theory where gravity, and hence the metric field and spatio-temporal distances between events, are subject to quantum mechanical laws. I will review the recent efforts of developing a framework to describe "superpositions of causal order", where one cannot say that A is before or after B, i.e. where the order between events is "indefinite". I will then discuss experimental perspectives for observing such indefinite causal structure and argue that they are useful resources for quantum information processing.