Systematics of the light curves of classical Cepheids with the longest known periods have been investigated with the help of full-amplitude models of pulsating stellar envelopes. For periods exceeding about 60 days, flat-topped light curves of the S Vul type are found to replace the smooth, asymmetric light curves characteristic of the slightly faster Cepheids. Predicted light and velocity amplitudes (although not the predicted radius amplitudes) agree well with observations. Variables with fluctuating light minima are observed to lie well off the mean period-luminosity relation, as are a few other (more stable?) variables with similarly long periods. The explanation for the long periods is probably low effective temperature rather than a low stellar mass. Because of the abnormal slowness of the classical Cepheids with periods longer than about 100 days, it is recommended that these variables not be used to calibrate the mean period-luminosity relation. Analogies between the slow classical Cepheids and the slow Population II Cepheids are drawn.