Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach starts with a large group $G$ and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from Kourovka Notebook.
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