Scientific Machine Learning (SciML) is an emerging scientific field at the interface of data science, high-dimensional approximation, statistics, and deep learning methodologies. Central themes are operator approximation and operator learning. This MS will present recent contributions on spectral, polynomial, and other "high-order'' approximation and computation methodologies for operator surrogates in SciML. Recent progress in spectral and polynomial approximation theoretical results and approximation algorithms will be reviewed, on the one hand in operator learning and on the other hand in deep learning and statistical learning theory and computation in relation to operator approximation.