A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grübler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grübler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grübler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. Firstly, the mobility research background and the Modified Grübler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grübler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grübler-Kutzbach criterion is demonstrated based on screw theory. The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grübler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grübler-Kutzbach criterion. A generally applicable criterion for all mechanism mobility has been an an active domain in mechanism theory lasting more than 150 years. It is said that the Modified Grübler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grübler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grübler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. and the Modified Grübler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grübler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grübler-Kutzbach The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grübler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grübler-Kutzbach criterion.