  We investigate the detailed properties of Observational entropy, introduced
by \v{S}afr\'{a}nek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization
of Boltzmann entropy to quantum mechanics. This quantity can involve multiple
coarse-grainings, even those that do not commute with each other, without
losing any of its properties. It is well-defined out of equilibrium, and for
some coarse-grainings it generically rises to the correct thermodynamic value
even in a genuinely isolated quantum system. The quantity contains several
other entropy definitions as special cases, it has interesting
information-theoretic interpretations, and mathematical properties -- such as
extensivity and upper and lower bounds -- suitable for an entropy. Here we
describe and provide proofs for many of its properties, discuss its
interpretation and connection to other quantities, and provide numerous
simulations and analytic arguments supporting the claims of its relationship to
thermodynamic entropy. This quantity may thus provide a clear and well-defined
foundation on which to build a satisfactory understanding of the second
thermodynamical law in quantum mechanics.
