Given integers r\geq 2 and n,t\geq 1 we call families \mathcal{F}_1,...,\mathcal{F}_r\subseteq 2^[n] r-cross t-intersecting if for all F_i in \mathcal{F}_i, i in [r], we have \bigcap_{i in [r]}F_i\geq t. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of \sum_{j in [r]} |\mathcal{F}_j| for r-cross t-intersecting non-empty families in the cases when these are k-uniform families (and n\geq 3k-t) or arbitrary subfamilies of 2^[n]. We obtain the aforementioned theorems as instances of a more general result that considers measures over the families. This also provides the maximum of \sum_{j in [r]}|\mathcal{F}_j| for families of possibly mixed uniformities k_1,...,k_r.
