In this talk I will discuss the novel experimental designs for large-scale multiple hypothesis testing problems. Testing to determine which genes are differentially expressed in a certain disease is a classic instance of multiple testing in medical informatics. Tremendous progress has been made in high-dimensional inference and testing problems by exploiting intrinsic low-dimensional structure. Sparsity is perhaps the simplest model for low-dimensional structure. It is based on the assumption that the signal of interest can be represented as a combination of a small number of elementary components. Sparse recovery is the problem of determining which components are needed in the representation based on measurements of the signal. For example, diseases are often characterized by a relatively small number of genes, which can be identified using high-throughput experimental techniques. This talk focuses on two issues related to this line of research. 1. Most theory and methods for sparse recovery are based on non-adaptive measurements. I will discuss the advantages of sequential measurement schemes that adaptively focus sensing using information gathered throughout the measurement process. In particular, I will show that sequential testing procedures can be significantly more powerful than non-sequential methods in the high-dimensional setting. 2. The standard sparse recovery problem involves inferring sparse linear functions. I will discuss generalizations of the standard problem to the recovery of sparse multilinear functions. Such functions are characterized by multiplicative interactions between the input variables, with sparsity meaning that relatively few of all conceivable interactions are present. This problem is motivated by the study of interactions between processes in complex networked systems (e.g., among genes and proteins in living cells). Our results extend the notion of compressed sensing from the linear sparsity model to nonlinear forms of sparsity encountered in complex systems. In contrast to linear sparsity models, in the multilinear case the pattern of sparsity can significantly affect sensing requirements.