Six lectures on translation-invariant operators and subspaces

This paper represents the text of a minicourse delivered by the author for participants of the "Workshop on Measure Theory and Real Analysis", Grado, September-97. We consider topics on multiplier theory and translation-invariant subspaces in function spaces on groups T and ℤ. Our goal is to give an introduction to the subject from the very beginning up to some recent results. The presentation is aimed at graduate students; no preliminaries in Fourier Analysis are supposed.