These are the lecture notes for a one semester course at Leibniz University Hannover. The main aim of the course is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model some materials in condensed matter physics or lattice gases. The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) short discussion on related topics for further reading. The level is aimed at students who have at least had some exposure to (functional) analysis and have a certain mathematical "maturity".