  Motivated by the qualitative picture of Canonical Typicality, we propose a
refined formulation of the Eigenstate Thermalization Hypothesis (ETH) for
chaotic quantum systems. The new formulation, which we refer to as subsystem
ETH, is in terms of the reduced density matrix of subsystems. This strong form
of ETH outlines the set of observables defined within the subsystem for which
it guarantees eigenstate thermalization. We discuss the limits when the size of
the subsystem is small or comparable to its complement. In the latter case we
outline the way to calculate the leading volume-proportional contribution to
the von Neumann and Renyi entanglment entropies. Finally, we provide numerical
evidence for the proposal in the case of a one-dimensional Ising spin-chain.
