In many practical settings one can sequentially and adaptively guide the collection of future data, based on information extracted from data already collected, in what is known as sequential experimental design, active learning, or adaptive sensing/sampling (depending on the context). The intricate relation between data analysis and acquisition in adaptive sensing paradigms is extremely powerful, and allows for reliable estimation in situations where non-adaptive sensing would fail dramatically. In this talk I consider estimation and detection of high-dimensional structured sparse signals in noise, and present (near)-optimal adaptive sensing procedures that provably outperform the best possible inference methods based on non-adaptive sensing. The methods developed can also be used also for adaptive compressive sensing and for detection of (sparse) correlations (this talk is based mainly on joint works with Ervin Tánczos, and with Pierre-André Savalle and Gábor Lugosi).