  The seventh part of the OGLE-III Catalog of Variable Stars (OIII-CVS)
consists of 4630 classical Cepheids in the Small Magellanic Cloud (SMC). The
sample includes 2626 fundamental-mode (F), 1644 first-overtone (1O), 83
second-overtone (2O), 59 double-mode F/1O, 215 double-mode 1O/2O, and three
triple-mode classical Cepheids. For each object basic parameters, multi-epoch
VI photometry collected within 8 or 13 years of observations, and finding
charts are provided in the OGLE Internet archive. We present objects of
particular interest: exceptionally numerous sample of single-mode
second-overtone pulsators, five double Cepheids, two Cepheids with eclipsing
variations superimposed on the pulsation light curves. At least 139
first-overtone Cepheids exhibit low-amplitude secondary variations with periods
in the range 0.60-0.65 of the primary ones. These stars populate three distinct
sequences in the Petersen diagram. The origin of this secondary modulation is
still unknown. Contrary to the Large Magellanic Cloud (LMC) we found only a few
candidates for anomalous Cepheids in the SMC. This fact may be a clue for the
explanation of the origin of the anomalous Cepheids. The period and luminosity
distributions of Cepheids in both Magellanic Clouds suggest that there are two
or three populations of classical Cepheids in each of the galaxies. The main
difference between the LMC and SMC lays in different numbers of Cepheids in
each group. We fit the period-luminosity (PL) relations of SMC Cepheids and
compare them with the LMC PL laws.
