This survey by David P. Woodruff highlights recent advances in numerical linear algebra using linear sketching techniques. Linear sketching involves compressing a given matrix by multiplying it with a random matrix to reduce its size, which allows for efficient computation on the smaller matrix. The survey covers several applications, including least squares regression, robust regression, low-rank approximation, and graph sparsification. It discusses various variants of these problems and the limitations of sketching methods. The focus is on over-constrained systems, where the number of observations is much larger than the number of predictor variables. The survey also delves into the theoretical foundations of subspace embeddings, matrix multiplication, and the role of high probability and leverage scores in these techniques. Key results include optimal algorithms for approximate least squares regression and the use of Johnson-Lindenstrauss transforms and sparse embedding matrices to achieve efficient sketching.This survey by David P. Woodruff highlights recent advances in numerical linear algebra using linear sketching techniques. Linear sketching involves compressing a given matrix by multiplying it with a random matrix to reduce its size, which allows for efficient computation on the smaller matrix. The survey covers several applications, including least squares regression, robust regression, low-rank approximation, and graph sparsification. It discusses various variants of these problems and the limitations of sketching methods. The focus is on over-constrained systems, where the number of observations is much larger than the number of predictor variables. The survey also delves into the theoretical foundations of subspace embeddings, matrix multiplication, and the role of high probability and leverage scores in these techniques. Key results include optimal algorithms for approximate least squares regression and the use of Johnson-Lindenstrauss transforms and sparse embedding matrices to achieve efficient sketching.