  Given two random variables $X$ and $Y$, an operational approach is undertaken
to quantify the ``leakage'' of information from $X$ to $Y$. The resulting
measure $\mathcal{L}(X \!\! \to \!\! Y)$ is called \emph{maximal leakage}, and
is defined as the multiplicative increase, upon observing $Y$, of the
probability of correctly guessing a randomized function of $X$, maximized over
all such randomized functions. A closed-form expression for $\mathcal{L}(X \!\!
\to \!\! Y)$ is given for discrete $X$ and $Y$, and it is subsequently
generalized to handle a large class of random variables. The resulting
properties are shown to be consistent with an axiomatic view of a leakage
measure, and the definition is shown to be robust to variations in the setup.
Moreover, a variant of the Shannon cipher system is studied, in which
performance of an encryption scheme is measured using maximal leakage. A
single-letter characterization of the optimal limit of (normalized) maximal
leakage is derived and asymptotically-optimal encryption schemes are
demonstrated. Furthermore, the sample complexity of estimating maximal leakage
from data is characterized up to subpolynomial factors. Finally, the
\emph{guessing} framework used to define maximal leakage is used to give
operational interpretations of commonly used leakage measures, such as Shannon
capacity, maximal correlation, and local differential privacy.
