Our first paper reviewed methods for modelling variation in cancer incidence and mortality rates in terms of either period effects or cohort effects in the general multiplicative risk model. There we drew attention to the difficulty of attributing regular trends to either period or cohort influences. In this paper we turn to the more realistic problem in which neither period nor cohort effects alone lead to an adequate description of the data. We describe the age-period-cohort model and show how its ambiguities surrounding regular trends 'intensify'. We recommend methods for presenting the results of analyses based upon this model which minimize the serious risk of misleading implications and critically review previous suggestions. The discussion is illustrated by an analysis of breast cancer mortality in Japan with special reference to the phenomenon of 'Clemmesen's hook'.