Darwinian evolution is an act of information processing: populations sense and measure the state of their environment and adapt by changing their configurations accordingly. Changes of the environment result in an irreversible out-of-equilibrium adaptive evolution, with a constant flow of information. Our goal is to understand the biological limits of information processing in evolving populations. We study a wide range of biological systems, including rapid evolution of viruses such as HIV, somatic evolution of cellular populations in the adaptive immune system of vertebrates, and adaptive evolution of gene regulation. Although distinct in many of their biological characteristics, we aim to identify common features in their biophysical principles, and ultimately to devise a common framework for a predictive description their evolutionary dynamics.