Age-period-cohort (APC) models are used to analyse age-specific disease or mortality rates provided for several periods in time with respect to three time scales: age, period (calendar period during which the incidence or mortality rates were observed) and cohort (time of birth). Frequently, several sets of such age-specific rates are observed because data were recorded according to one further stratification variable resulting in one set of rates for each stratum of this variable. For example, rates might be available for males and females or for several geographical regions. Each set of rates could be analysed separately by means of an univariate APC model. However, because of similar relevant risk factors it might be beneficial to analyse all sets of rates jointly treating some sets of time effects as common across strata. Multivariate APC models share sets of time effects, for example the age effects, while the remaining parameters can be different. This dissertation aims at improving the methodology for statistical inference in multivariate APC models. We first show that differences of stratum-specific time effects in multivariate APC models are identifiable, so that the well known identifiability problem for univariate APC models is avoided. We develop a multivariate Bayesian APC model based on smoothing priors to analyse heterogeneous time trends. This approach represents an attractive alternative to maximum likelihood (ML) based approaches when age groups and periods are given for the same time-interval widths and avoids the artefacts, e.g. artificial cyclical patterns, which occur in the case of unequal timeinterval widths. Subsequently, we present a conditional approach for inference in multivariate APC models. In contrast to the unconditional approach which includes many nuisance parameters, the conditional approach directly models the parameters of interest, namely the differences of stratumspecific time effects. Furthermore, we extend this approach to analyse datasets with multiple stratification factors. ML estimation is performed using software for multinomial logistic regression. The use of cubic smoothing splines is proposed to avoid artificial cyclical patterns in the case of unequally spaced time-intervals of age and period. Finally, we propose the use of correlated smoothing priors and correlated overdispersion parameters to capture the potential dependence present between multiple health outcomes. By means of case studies we demonstrate that correlated multivariate APC models are useful to improve the precision of relative risk estimates and to extrapolate missing data. We implement the methodology using Markov chain Monte Carlo (MCMC) and the recently proposed integrated nested Laplace approximation (INLA). With INLA it is possible to correlate a wide range of other latent Gaussian models, e.g. conditionally autoregressive models or seasonal models. In an application to Swiss suicide data from 1950 to 2007, we analyse gender-specific differences using both ordinary and correlated multivariate Bayesian APC models. Results indicate that males have approximately three times the risk of committing suicide as women. Elderly men and those between 15 − 24 are especially at risk. Furthermore, we use univariate and multivariate APC models to investigate whether explanatory variables related to family integration can explain gender-specific suicide trends.