  The JWST Advanced Deep Extragalactic Survey (JADES) is a multi-cycle JWST
program that has taken among the deepest near-/mid-infrared images to date
(down to $\sim$30 ABmag) over $\sim$25 arcmin$^2$ in the GOODS-S field in two
sets of observations with one year of separation. This presented the first
opportunity to systematically search for transients, mostly supernovae (SNe),
out to $z$$>$2. We found 79 SNe: 38 at $z$$<$2, 23 at 2$<$$z$$<$3, 8 at
3$<$$z$$<$4, 7 at 4$<$$z$$<$5, and 3 with undetermined redshifts, where the
redshifts are predominantly based on spectroscopic or highly reliable JADES
photometric redshifts of the host galaxies. At this depth, the detection rate
is $\sim$1-2 per arcmin$^2$ per year, demonstrating the power of JWST as a
supernova discovery machine. We also conducted multi-band follow-up NIRCam
observations of a subset of the SNe to better constrain their light curves and
classify their types. Here, we present the survey, sample, search parameters,
spectral energy distributions (SEDs), light curves, and classifications. Even
at $z$$\geq$2, the NIRCam data quality is high enough to allow SN
classification via multi-epoch light-curve fitting with confidence. The
multi-epoch SN sample includes a Type Ia SN at $z_{\mathrm{spec}}$$=$2.90, Type
IIP SN at $z_{\mathrm{spec}}$$=$3.61, and a Type Ic-BL SN at
$z_{\mathrm{spec}}$$=$2.845. We also found that two $z$$\sim$16 galaxy
candidates from the first imaging epoch were actually transients that faded in
the second epoch, illustrating the possibility that moderate/high-redshift SNe
could mimic high-redshift dropout galaxies.
