Interfolding machine for making an interfolded product starting from a web of paper, or similar material, comprising: - a feeding means adapted to feed at least one web of paper; - a cutting and transferring means (2,2',3,5) of said web of paper adapted to cut said web of paper into sheets of determined length and to transfer them along a conveying path; - an actuating means of said cutting and transferring means (2,2',3,5) that is arranged to cause said cutting and transferring means (2,2',3,5) to convey said web of paper and said sheets proceed at a first speed V1; - an overlapping means adapted to pick up said sheets from said conveying path and to overlap them in turn by a predetermined sheet portion; - an actuating means of said overlapping means adapted to cause said overlapping means to convey said sheets at a second speed V2 less than V1, said speed difference being imposed for overlapping partially two consecutive sheets according to said sheet portion; - a folding means (8,9) of the consecutive overlapped sheets into panels arranged to provide a stack of interfolded sheets; wherein said cutting and transferring means (2,2',3,5) comprises a first cutting, transferring and overlapping means for sheets of a first length and at least a second cutting, transferring and overlapping means for sheets of a second length characterised in that - said cutting and transferring means (2,2',3,5) comprises a transfer roller (5,3,2') having a first plurality of sheet head gripping means (27a,27b,27c,27d), which is arranged peripherally at a first angular distance and that is adapted to transfer to a sheet overlapping roller constituting said overlapping means said sheets of said first length, and a second plurality of sheet head gripping means (27'a,27'b,27'c), which is arranged peripherally at a second angular distance and that is adapted to transfer to said sheet overlapping roller said sheets of said second length; and said transfer roller (5,3,2') comprises furthermore, means for gripping and overlapping (29, 50) each sheet at least at a respective sheet tail such that a successive sheet that moves along said conveying path overlaps by said measured portion under a previous sheet whose head has been already gripped by said sheet overlapping roller and whose tail has been held by said means for gripping and overlapping (29, 50). - a feeding means adapted to feed at least one web of paper; - a cutting and transferring means (2,2',3,5) of said web of paper adapted to cut said web of paper into sheets of determined length and to transfer them along a conveying path; - an actuating means of said cutting and transferring means (2,2',3,5) that is arranged to cause said cutting and transferring means (2,2',3,5) to convey said web of paper and said sheets proceed at a first speed V1; - an overlapping means adapted to pick up said sheets from said conveying path and to overlap them in turn by a predetermined sheet portion; - an actuating means of said overlapping means adapted to cause said overlapping means to convey said sheets at a second speed V2 less than V1, said speed difference being imposed for overlapping partially two consecutive sheets according to said sheet portion; - a folding means (8,9) of the consecutive overlapped sheets into panels arranged to provide a stack of interfolded sheets; - said cutting and transferring means (2,2',3,5) comprises a transfer roller (5,3,2') having a first plurality of sheet head gripping means (27a,27b,27c,27d), which is arranged peripherally at a first angular distance and that is adapted to transfer to a sheet overlapping roller constituting said overlapping means said sheets of said first length, and a second plurality of sheet head gripping means (27'a,27'b,27'c), which is arranged peripherally at a second angular distance and that is adapted to transfer to said sheet overlapping roller said sheets of said second length; and