Universally Baire sets in $2^{\kappa}$

We generalize the basic theory of universally Baire sets of $2^\omega$ to a theory of universally Baire subsets of $2^\kappa$. We show that the fundamental characterizations of the property of being universally Baire have natural generalizations that can be formulated also for subsets of $2^\kappa$, in particular we provide four equivalent uniform definitions in the parameter $\kappa$ (for $\kappa$ an infinite cardinal) characterizing for each such $\kappa$ the class of universally Baire subsets of $2^\kappa$. For $\kappa=\omega$, these definitions bring us back to the original notion of universally Baire sets of reals given by Feng, Magidor and Woodin [2].