  Two decades ago, empirical evidence concerning the existence and frequency of
planets around stars, other than our own, was absent. Since this time, the
detection of extrasolar planets from Jupiter-sized to most recently Earth-sized
worlds has blossomed and we are finally able to shed light on the plurality of
Earth-like, habitable planets in the cosmos. Extrasolar moons may also be
frequent habitable worlds but their detection or even systematic pursuit
remains lacking in the current literature. Here, we present a description of
the first systematic search for extrasolar moons as part of a new observational
project called "The Hunt for Exomoons with Kepler" (HEK). The HEK project
distills the entire list of known transiting planet candidates found by Kepler
(2326 at the time of writing) down to the most promising candidates for hosting
a moon. Selected targets are fitted using a multimodal nested sampling
algorithm coupled with a planet-with-moon light curve modelling routine. By
comparing the Bayesian evidence of a planet-only model to that of a
planet-with-moon, the detection process is handled in a Bayesian framework. In
the case of null detections, upper limits derived from posteriors marginalised
over the entire prior volume will be provided to inform the frequency of large
moons around viable planetary hosts, eta-moon. After discussing our
methodologies for target selection, modelling, fitting and vetting, we provide
two example analyses.
