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　Grubler-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　Grubler-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　Grubler-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.