Aluminosilicate glasses are of great geological and technological importance. Significant efforts have been spent for enhancing the insight into their structures, where magic-angle-spinning (MAS) NMR that exploits the spin-5/2 27Al as probe nucleus constitutes one widely utilized option. We review the application of basic 27Al NMR experimentation for studying primarily the short-range (less than or similar to 0.3 nm) structure of aluminosilicate glasses, emphasizing practical aspects of performing MAS and triple-quantum MAS NMR experiments, as well as options for data analysis to extract 27Al NMR parameters and quantifying AlOp populations. We illustrate the strengths and weaknesses of routine 27Al NMR for investigating aluminosilicate glasses, including its development over time. While parts of the text apply generally to MAS NMR targeting half-integer spins as structural probes in crystalline as well as amorphous materials, the focus remains on 27Al NMR applications to aluminosilicate glasses, whose basic structural features are outlined together with a survey of the most central research problems in the field. By providing both in-depth discussions about the building blocks of aluminosilicate glasses while assuming a modest background knowledge of the reader about MAS NMR and glass structure, we hope that the presentation will appeal to a broad audience, encompassing both experienced researchers in solid-state NMR or glass structures, as well as to beginners in either area.