  This paper studies forward and reverse projections for the R\'{e}nyi
divergence of order $\alpha \in (0, \infty)$ on $\alpha$-convex sets. The
forward projection on such a set is motivated by some works of Tsallis {\em et
al.} in statistical physics, and the reverse projection is motivated by robust
statistics. In a recent work, van Erven and Harremo\"es proved a Pythagorean
inequality for R\'{e}nyi divergences on $\alpha$-convex sets under the
assumption that the forward projection exists. Continuing this study, a
sufficient condition for the existence of forward projection is proved for
probability measures on a general alphabet. For $\alpha \in (1, \infty)$, the
proof relies on a new Apollonius theorem for the Hellinger divergence, and for
$\alpha \in (0,1)$, the proof relies on the Banach-Alaoglu theorem from
functional analysis. Further projection results are then obtained in the finite
alphabet setting. These include a projection theorem on a specific
$\alpha$-convex set, which is termed an {\em $\alpha$-linear family},
generalizing a result by Csisz\'ar for $\alpha \neq 1$. The solution to this
problem yields a parametric family of probability measures which turns out to
be an extension of the exponential family, and it is termed an {\em
$\alpha$-exponential family}. An orthogonality relationship between the
$\alpha$-exponential and $\alpha$-linear families is established, and it is
used to turn the reverse projection on an $\alpha$-exponential family into a
forward projection on a $\alpha$-linear family. This paper also proves a
convergence result of an iterative procedure used to calculate the forward
projection on an intersection of a finite number of $\alpha$-linear families.
