Several methods are available to bound the norm of random matrices with independent entries. A more recent line of work studies sums of independent random matrices. I will present two recent results on the norm of 'non-standard' random matrices, that do not fit in the above framework. These models arise from applications to statistics and computer science, more precisely: 1) Analysis of semidefinite programming hierarchies in combinatorial optimization; 2) Algorithms for sparse covariance estimation in high-dimensional statistics.