We propose new summary statistics quantifying several forms of dependence between points of different types in a multi-type spatial point pattern. These statistics are the multivariate counterparts of the J-function for point processes of a single type, introduced by Lieshout & Baddeley (1996). They are based on comparing distances from a type i point to either the nearest type j point or to the nearest point in the pattern regardless of type to these distances seen from an arbitrary point in space. Information about the range of interaction can also be inferred. Our statistics can be computed explicitly for a range of well-known multivariate point process models. Some applications to bivariate and trivariate data sets are presented as well.