We introduce an information-theoretic method for quantifying causality in chaotic systems. The approach, referred to as IT-causality, quantifies causality by measuring the information gained about future events conditioned on the knowledge of past events. The causal interactions are classified into redundant, unique, and synergistic contributions depending on their nature. The formulation is non-intrusive, invariance under invertible transformations of the variables, and provides the missing causality due to unobserved variables. The method only requires pairs of past-future events of the quantities of interest, making it convenient for both computational simulations and experimental investigations. IT-causality is validated in four scenarios representing basic causal interactions among variables: mediator, confounder, redundant collider, and synergistic collider. The approach is leveraged to address two questions relevant to turbulence research: i) the scale locality of the energy cascade in isotropic turbulence, and ii) the interactions between inner and outer layer flow motions in wall-bounded turbulence. In the former case, we demonstrate that causality in the energy cascade flows sequentially from larger to smaller scales without requiring intermediate scales. Conversely, the flow of information from small to large scales is shown to be redundant. In the second problem, we observe a unidirectional causality flow, with causality predominantly originating from the outer layer and propagating towards the inner layer, but not vice versa. The decomposition of IT-causality into intensities also reveals that the causality is primarily associated with high-velocity streaks.