The nature of dark matter is a longstanding enigma of physics; it may consist of particles beyond the Standard Model that are still elusive to experiments. Among indirect search techniques, which look for stable products from the annihilation or decay of dark matter particles, or from axions coupling to high-energy photons, observations of the $gamma$-ray sky have come to prominence over the last few years, because of the excellent sensitivity of the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope mission. The LAT energy range from 20 MeV to above 300 GeV is particularly well suited for searching for products of the interactions of dark matter particles. In this report we describe methods used to search for evidence of dark matter with the LAT, and review the status of searches performed with up to six years of LAT data. We also discuss the factors that determine the sensitivities of these searches, including the magnitudes of the signals and the relevant backgrounds, considering both statistical and systematic uncertainties. We project the expected sensitivities of each search method for 10 and 15 years of LAT data taking. In particular, we find that the sensitivity of searches targeting dwarf galaxies, which provide the best limits currently, will improve faster than the square root of observing time. Current LAT limits for dwarf galaxies using six years of data reach the thermal relic level for masses up to 120 GeV for the $bbar{b}$ annihilation channel for reasonable dark matter density profiles. With projected discoveries of additional dwarfs, these limits could extend to about 250 GeV. With as much as 15 years of LAT data these searches would be sensitive to dark matter annihilations at the thermal relic cross section for masses to greater than 400 GeV (200 GeV) in the $bbar{b}$ ($tau^+ tau^-$) annihilation channels.
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