  The conventional channel resolvability problem refers to the determination of
the minimum rate required for an input process so that the output distribution
approximates a target distribution in either the total variation distance or
the relative entropy. In contrast to previous works, in this paper, we use the
(normalized or unnormalized) R\'enyi divergence (with the R\'enyi parameter in
$[0,2]\cup\{\infty\}$) to measure the level of approximation. We also provide
asymptotic expressions for normalized R\'enyi divergence when the R\'enyi
parameter is larger than or equal to $1$ as well as (lower and upper) bounds
for the case when the same parameter is smaller than $1$. We characterize the
R\'enyi resolvability, which is defined as the minimum rate required to ensure
that the R\'enyi divergence vanishes asymptotically. The R\'enyi
resolvabilities are the same for both the normalized and unnormalized
divergence cases. In addition, when the R\'enyi parameter smaller than~$1$,
consistent with the traditional case where the R\'enyi parameter is equal
to~$1$, the R\'enyi resolvability equals the minimum mutual information over
all input distributions that induce the target output distribution. When the
R\'enyi parameter is larger than $1$ the R\'enyi resolvability is, in general,
larger than the mutual information. The optimal R\'enyi divergence is proven to
vanish at least exponentially fast for both of these two cases, as long as the
code rate is larger than the R\'enyi resolvability. The optimal exponential
rate of decay for i.i.d.\ random codes is also characterized exactly. We apply
these results to the wiretap channel, and completely characterize the optimal
tradeoff between the rates of the secret and non-secret messages when the
leakage measure is given by the (unnormalized) R\'enyi divergence.
