The first topics of study in many courses in graph theory are matchings, paths, and cycles in graphs. We introduce basic graph theoretic methods and focus on extremal aspects of these topics. We start with k-cores and maximum cuts in graphs, and move on to Hall’s Theorem and Tutte’s 1-Factor Theorem, the Tutte–Berge Formula, and the extremal problem for matchings. We study the extremal problem for paths in graphs, covering the Erdős–Gallai Theorem and the Erdős–Sós Conjecture, Perles’ proof via geometric graphs, Kopylov’s Theorem, the Pósa’s Rotation Lemma, and theorems on hamiltonian graphs such as Dirac’s Theorem and the Bondy–Chvátal closure theorem, and the Smith–Thomason theorem on uniquely hamiltonian graphs.