From an operational perspective, quantum theory can be understood as a set of rules that prescribe probabilities for the outcomes of experiments composed in different configurations. In its standard form, however, this understanding presumes a notion of background time. In the theory of general relativity, the causal structure of space-time is dynamical and not predefined, which suggests that a quantum theory of gravity may require a more general operational paradigm. In this talk, I will describe recent progress in this direction. First, I will show how assuming only the local validity of standard quantum mechanics, but relaxing the assumption that local operations take place in a global causal structure, leads us to a generalized framework that unifies all signaling and non-signaling quantum correlations in space-time via an extension of the density matrix called the process matrix. This framework also contains the possibility for a new type of correlations, which are incompatible with any underlying causal structure. Such correlations violate causal inequalities, whose general theory I will describe. I will then present an extension of the process matrix framework, in which no predefined causal structure is assumed even locally. This is based on a more general, time-neutral notion of operation, which provides new insights into the problem of time-reversal symmetry in quantum mechanics, the meaning of causality, and the fact that we can remember the past but not the future. In the resultant generalized formulation of quantum theory, operations are associated with regions that can be connected in networks through their boundaries without prior directionality assumed for the connections. The theory is compatible with timelike loops and other acausal structures. \\ Based on: ...