  We present the results of a survey aimed at discovering and studying
transiting planets with orbital periods shorter than one day
(ultra--short-period, or USP, planets), using data from the {\em Kepler}
spacecraft. We computed Fourier transforms of the photometric time series for
all 200,000 target stars, and detected transit signals based on the presence of
regularly spaced sharp peaks in the Fourier spectrum. We present a list of 106
USP candidates, of which 18 have not previously been described in the
literature. In addition, among the objects we studied, there are 26 USP
candidates that had been previously reported in the literature which do not
pass our various tests. All 106 of our candidates have passed several standard
tests to rule out false positives due to eclipsing stellar systems. A low false
positive rate is also implied by the relatively high fraction of candidates for
which more than one transiting planet signal was detected. By assuming these
multi-transit candidates represent coplanar multi-planet systems, we are able
to infer that the USP planets are typically accompanied by other planets with
periods in the range 1-50 days, in contrast with hot Jupiters which very rarely
have companions in that same period range. Another clear pattern is that almost
all USP planets are smaller than 2 $R_\oplus$, possibly because gas giants in
very tight orbits would lose their atmospheres by photoevaporation when subject
to extremely strong stellar irradiation. Based on our survey statistics, USP
planets exist around approximately $(0.51\pm 0.07)\%$ of G-dwarf stars, and
$(0.83\pm 0.18)\%$ of K-dwarf stars.
