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 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 that is arranged to cause said cutting and transferring means 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 of the consecutive overlapped sheets into panels arranged to provide a stack of interfolded sheets ; wherein said cutting and transferring means 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 comprises a transfer roller having a first plurality of sheet head gripping means , 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 comprises furthermore, means for gripping and overlapping 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 . - a feeding means adapted to feed at least one web of paper; - a cutting and transferring means 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 that is arranged to cause said cutting and transferring means 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 of the consecutive overlapped sheets into panels arranged to provide a stack of interfolded sheets ; - said cutting and transferring means comprises a transfer roller having a first plurality of sheet head gripping means , 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 <EOS>