Abstract: What probabilistic phenomena can arise when a random system is conditioned on partial information? Such questions are of fundamental interest to the understanding of conditional distributions, and at the same time have implications for applications ranging from GPS navigation to weather forecasting. A general understanding of classical problems of this kind, dating back at least to Blackwell (1957), has only recently been obtained. At the same time, new questions and phenomena that arise in high-dimensional problems, such as conditional phase transitions and decay of correlations, are only beginning to be addressed. In this talk I will aim to give an overview of some old and new problems in this area, with particular emphasis on the behavior of conditional distributions in high dimension and its implications for the design and analysis of algorithms for data assimilation in complex systems.