The goal of an investigation, scientific or otherwise, is usually to find answers to some specific set of questions about the state of nature: what is the seismic velocity structure? How likely is this volcano to erupt within a certain period? Does a subsurface reservoir contain resources of interest? Background research may reveal the existence of pertinent knowledge and information discovered previously, new data are normally acquired, and an inference problem is solved in order to answer the questions taking both all of the a priori information and the new data into account. Inverse theory, decision theory and the theory of experimental design provide methods to optimize the design of the investigation and to estimate results. However, those theories are normally set in the context of a particular model of the universe, with its particular parametrization. This requires the investigator to specify a priori a coherent utility (a function that describes the risks and rewards) of all possible outcomes under that parametrization. Quite commonly, the investigator may not be able to do this. Ideally an investigator would be able merely to pose a set of questions, define a set of constraints on the data types, acquisition costs and logistics, and provide a functional to relate the questions to any particular parameter space. Theory and methodology would then semi-autonomously drive the interrogation of the state of nature by optimally selecting one or more relevant models and parameter spaces, and designing, acquiring and analysing data, in order to best answer the questions. If necessary this could be done in a sequential or iterative manner, which potentially then involves changing the questions posed in each iteration based on both previous results and inspiration from the investigator. We present such a theory of interrogation in this paper. We review the relevant aspects of decision and design theory, and cast them in a framework where the investigator specifies a utility only at the level required by the general questions to be posed. Each model under consideration is then mapped into this utility space of possible answers. We then extend this framework to sequential investigations, where the outcome of each step may affect all aspects of the problem: the models entertained, the utilities and even the questions themselves. A variety of examples illustrates the generality of this method: an asset team investigating how best to exploit a subsurface reservoir, Monte Carlo sampling to estimate the Bayesian evidence for geophysical models, discriminating between different rock physics models of strain in laboratory deformation experiments, an organization sequentially assessing the effectiveness of its methods to evaluate subsurface assets, assessing whether subsurface CO₂ storage should be promoted for climate change mitigation, and examples running through the text of seismic tomography, earthquake characterization and autonomous interplanetary robotic exploration.