Stirring of water by mesoscale currents (“eddies”) leads to large-scale transport of many important oceanic properties (“tracers”). These eddy-induced transports can be related to the large-scale tracer gradients, using the concept of turbulent diffusion. The concept is widely used to describe these transports in the real ocean and to represent them in climate models. This study focuses on the inherent complexity of the corresponding eddy diffusivity tensor, defined here in all its spatio-temporal complexity. Results demonstrate that this comprehensive diffusivity tensor is space-, time-, direction- and tracer-dependent. Using numerical simulations with both idealized and comprehensive models of the Atlantic circulation, we show that these properties lead to upgradient eddy fluxes and the potential importance of all tensor components. Implications of all this complexity for the development of eddy parameterization schemes in climate models and diffusivity estimates are discussed.