Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties In this volume the authors construct a theory of weights on the log crystalline cohomologies of families of open smooth varieties in characteristic p>0 by defining and constructing four filtered complexes. Fundamental properties of these filtered complexes are proved in particular the p adic purity the functionality of three filtered complexes the weight filtered base change formula the weight filtered Künneth formula the weight filtered Poincaré duality and the E2 degeneration of p adic weight spectral sequences. In addition the authors state some theorems on the weight filtration and the slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic p>0. Preliminaries on Filtered Derived Categories and Topoi Weight Filtrations on Log Crystalline Cohomologies Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary).