Motivated by the Frankl's results in [P. Frankl, Multiply-intersecting families, J. Combin. Theory B 53 (1991) 195–234], we consider some problems concerning the maximum size of multiply-intersecting families with additional conditions. Among other results, we show the following version of the Erdős–Ko–Rado theorem: for all and there exist positive constants ε and such that if and then r-wise t-intersecting k-uniform families on n vertices have size at most .