  We present a search for extragalactic fast blue optical transients (FBOTs)
during Phase I of the Zwicky Transient Facility (ZTF). We identify 38
candidates with durations above half-maximum light 1 d < t1/2 < 12 d, of which
28 have blue (g-r<-0.2 mag) colors at peak light. Of the 38 transients (28
FBOTs), 19 (13) can be spectroscopically classified as core-collapse supernovae
(SNe): 11 (8) H- or He-rich (Type II/IIb/Ib) SNe, 6 (4) interacting (Type
IIn/Ibn) SNe, and 2 (1) H&He-poor (Type Ic/Ic-BL) SNe. Two FBOTs (published
previously) had high-S/N predominantly featureless spectra and luminous radio
emission: AT2018lug and AT2020xnd. Seven (five) did not have a definitive
classification: AT 2020bdh showed tentative broad H$\alpha$ in emission, and AT
2020bot showed unidentified broad features and was 10 kpc offset from the
center of an early-type galaxy. Ten (six) have no spectroscopic observations or
redshift measurements. We present multiwavelength (radio, millimeter, and/or
X-ray) observations for five FBOTs (three Type Ibn, one Type IIn/Ibn, one Type
IIb). Additionally, we search radio-survey (VLA and ASKAP) data to set limits
on the presence of radio emission for 22 of the transients. All X-ray and radio
observations resulted in non-detections; we rule out AT2018cow-like X-ray and
radio behavior for five FBOTs and more luminous emission (such as that seen in
the Camel) for four additional FBOTs. We conclude that exotic transients
similar to AT2018cow, the Koala, and the Camel represent a rare subset of
FBOTs, and use ZTF's SN classification experiments to measure the rate to be at
most 0.1% of the local core-collapse SN rate.
