We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent $n_0$ such that for every odd integer $n≥ n_0$, the well-known classical $C’(1/6)$-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite $n$-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of $n$-periodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of $n$-periodic groups. As applications, we show the undecidability of Markov properties in classes of $n$-periodic groups, we produce $n$-periodic groups whose Cayley graph contains an embedded expander graphs, and we give an $n$-periodic version of the Rips construction. We also obtain simpler proofs of some known results like the existence of uncountably many finitely generated $n$-periodic groups and the SQ-universality (in the class of $n$-periodic groups) of free Burnside groups.