Following its inception in the mid-19th century, our understanding of thermodynamic entropy has undergone many revisions, most notably through the development of microscopic descriptions by Boltzmann and Gibbs, which led to a deep understanding of equilibrium thermodynamics. The role of entropy has since moved beyond the traditional boundaries of equilibrium thermodynamics, towards problems for which the development of a statistical mechanical theory seems plausible but the a-priori probabilities of states are not known, making the definition and calculation of entropy-like quantities challenging. In this talk, we will discuss two new classes of methods that enable these computations: one based on pattern matching ideas from information theory, and the other based on basin volume calculations. These approaches provide us with very general frameworks for computing entropy, density of states, and entropy production in systems far from equilibrium. We will discuss applications of these ideas to a variety of contexts: from granular systems, to absorbing-state models, to active matter, in simulations and in experiments. Throughout the talk, I will highlight challenges and promising future directions for these measurements.