Seismic reservoir characterization focuses on the prediction of reservoir properties based on the available geophysical and petrophysical data. The inverse problem generally includes continuous properties, such as petrophysical and elastic attributes, and discrete properties, such as lithology/fluid classes. We have developed a joint probabilistic inversion methodology for the prediction of petrophysical and elastic properties and lithology/fluid classes that combined statistical rock physics and Bayesian seismic inversion. The elastic attributes depend on continuous petrophysical variables, such as porosity and clay content, and discrete lithology/fluid classes, through a nonlinear rock-physics relationship together. The seismic model relates the elastic attributes, such as velocities and density, to their seismic response (reflectivity, traveltime, and amplitudes). The advantage of our integrated approach is that the inversion method accounts for the uncertainty associated to each step of the modeling workflow. The lithology/fluid classes are assigned by a Markov random field prior model to capture vertical continuity and vertical sorting of the lithology/fluid classes. Because rock and fluid properties are in general not Gaussian, a spatially coupled Gaussian mixture prior model based on the lithology/fluid classes is constructed. The forward geophysical operator includes a lithology-/fluid-dependent rock physics model and a linearized seismic model based on the convolution of the seismic wavelet with the reflectivity coefficient series. The solution of the inverse problem consists of the posterior distributions of petrophysical and elastic properties and lithology/fluid classes. We proposed an efficient Markov chain Monte Carlo algorithm to sample from the posterior models and assess the uncertainty. Our methodology is demonstrated on a seismic cross section from a survey in the Norwegian Sea, and it shows promising results consistent with well-log data measured at the well location as well as reliable prediction uncertainties.