  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 $n\geq n_0$, 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.
