Abstract: Mutual information, defined via the linear combinations of von Neumann entanglement entropies, measures the total correlations between two subsystems and has played a vital role in characterizing equilibrium and non-equilibrium phenomena in many-body physics. In practice, its Renyi counterpart is usually computed owing to its simplicity. However, this commonly used notion of Renyi Mutual Information (RMI) cannot be interpreted as a measure of correlation, because it is neither non-negative nor does it satisfy the Data Processing Inequality. In this talk, we will explore several classes of properly defined RMIs. In particular, we develop an easily implementable replica trick which enables us to evaluate these RMIs in a variety of many-body systems, including 2D conformal field theories and random tensor networks. We will also discuss the prospect of measuring these new RMIs in quantum simulator experiments.