We present recently developed iterative methods for approximating matrices and tensors with structural constraints such as rank and sparsity level and pattern, extending to settings where data is incomplete or indirectly observed, with or without noise. These methods aim to solve canonical problems of numerical (multi-)linear algebra, namely approximate matrix inversion and low-rank matrix and tensor approximation, when structural constraints are imposed on the approximation. While the presented results revolve around aspects of sparsity, if time permits, we extend the presentation to other structural features such as non-negativity. We also present how this work contributes to the ongoing development of the repositories ApproximateMatrixInverses.jl, StructuredLowRankMatrices.jl, and StructuredLowRankTensors.jl.