A first stream of non-overlapping spaced-apart sheets which move at a high speed is converted into a second stream of partially overlapping sheets by braking successive sheets of the first stream and deflecting the trailing portions of braked sheets laterally to enable the oncoming foremost sheets of the first stream to catch up with and partly overlap the preceding sheets. When the first stream exhibits a gap as a result of the absence of one or more sheets, or independently of the presence or absence of gaps, the deflection of the braked sheet is repeated at a location which is nearer to the braking station to thus insure that the trailing portion of the braked sheet preceding a gap is deflected laterally immediately before the sheet which follows the gap catches up therewith. The deflection of trailing portions which precede gaps in the first stream is effected by suction. The locus of such deflection can be shifted lengthwise of the path for the sheets in dependency on changes of the ratio of the speeds of first and second streams, on changes of the width of clearances between the sheets of the first stream and/or in dependency on changes of the length of sheets.