Developing a general mobility method/formula is a hot topic lasting for more than 150 years in kinematics. It is necessary to apply any mobility method to puzzling overconstrained mechanisms for verification of its generality. Altmann linkages are such recognized puzzling mechanisms that their mobility analysis is of important significance. A necessary condition for judging a general mobility method is that the method can be fit for Altmann linkages. Firstly, this study classes Altmann linkages into 17 types in terms of the numbers and types of kinematic pairs, and then Altmann overconstrained linkages are further classified into 4 types. Secondly, the mobility of Altmann overconstrained linkages is systematically analyzed by the Modified Grübler-Kutzbach criterion based on screw theory, where passive freedoms are defined as limb passive freedoms and mechanism passive freedoms. In addition, the full-cycle mobility is judged, which overcomes the shortcoming of instantaneous property of screw theory. It is shown that Modified Grübler-Kutzbach criterion not only obtains the correct numerical mobility, but also gives the mobility character by resolving reciprocal screws for the constraint system. This study lays the foundation of verification for the generality of Modified Grübler-Kutzbach criterion. Besides, Altmann overconstrained linkages almost comprise all kinds of modern parallel mechanisms and some classical mechanisms, which provides an important reference for mechanism mobility calculation. Developing a general mobility method / formula is a hot topic lasting longer than 150 years in kinematics. It is necessary to apply any mobility method to puzzling over concavity mechanisms for verification of its generality. Altmann links are such recognized puzzling mechanisms that their mobility analysis is First, this study classes Altmann linkages into 17 types in terms of numbers and types of kinematic pairs, and then Altmann overconstrained linkages. are further classified into 4 types. Secondly, the mobility of Altmann overconstrained linkages is systematically analyzed by the Modified Grübler-Kutzbach criterion based on screw theory, where passive freedoms are defined as limb passive freedoms and mechanism passive freedoms. In addition, the full- cycle mobility is judged, which overcomes the shortcoming of instantaneous property of screw theory. It is shown that Modified Grübler-Kutzbach criterion not only obtains the correct numerical mobility, but also gives the mobility character by resolving reciprocal screws for the constraint system. This study lays the foundation of verification for the generality of Modified Grübler-Kutzbach criterion. Besides, Altmann overconstrained linkages almost include all kinds of modern parallel mechanisms and some classical mechanisms, which provide an important reference for mechanism mobility calculation.