Much of the recent progress in understanding quantum theory has been achieved within an operational approach. Within this context quantum mechanics is viewed as a theory for making probabilistic predictions for measurement outcomes following specified preparations. However, thus far some of the essential elements of the theory Ã¢ÂÂ space, time and causal structure Ã¢ÂÂ elude such an operational formulation and are assumed to be fixed. Is it possible to extend the operational approach to quantum mechanics such that the notions of an underlying spacetime or causal structure are not assumed? What new phenomenology can follow from such an approach? We develop a framework for multipartite quantum correlations that does not presume these notions, but simply that experimenters in their local laboratories are free to perform arbitrary quantum operations. All known situations that respect definite causal order, including signalling and no-signalling correlations between space-like and time-like separated experiments, as well as probabilistic mixtures of these, can be expressed in this framework. Remarkably, we find quantum correlations which are neither causally ordered nor in a probabilistic mixture of definite causal orders. These correlations are shown to enable performing a communication task that is impossible if a fixed background time is assumed and the events are sufficiently localized in the time.