The paper generalizes the classical theorem of A. J. W. Hilton and E. C. Milner [Q. J. Math., Oxf. II. Ser. 18, 369–384 (1967; Zbl 0168.26205)] for cross-intersecting families in various directions. Let \(n\), \(r\) and \(t\) be positive integers, where \(r \geq 2\). If \(\mathcal{F}_1, \dots, \mathcal{F}_r\) are families of sets such that \(|\bigcap_{i=1}^r F_i| \geq t\) for every \((F_1, \dots, F_r) \in \mathcal{F}_1 \times \dots \times \mathcal{F}_r\), then they are said to be \(r\)-cross \(t\)-intersecting. The authors determine the maximum sum of sizes of \(r\) non-empty \(r\)-cross \(t\)-intersecting families \(\mathcal{F}_1, \dots, \mathcal{F}_r\) of subsets of \([n] = \{1, \dots, n\}\). They also address the more general setting where \(\mathcal{F}_1, \dots, \mathcal{F}_r\) are equipped with certain natural and important measures, such as the product measure and the uniform measure. For each \(i \in [r]\), let \(\mu_i \colon \{0, 1, \dots, n\} \rightarrow \mathbb{R}_{\geq 0}\), and let \(\mu_i(\mathcal{F}_i)\) denote the measure \(\sum_{F \in \mathcal{F}_i} \mu_i(|F|)\) of \(\mathcal{F}_i\). The authors determine the largest possible value of \(\sum_{i=1}^r \mu_i(\mathcal{F}_i)\) for the case where \(\mu_1, \dots, \mu_r\) are non-increasing. For \(1 \leq k_1 \leq \cdots \leq k_r \leq n\), they determine the largest possible value of \(\sum_{i=1}^r \mu_i(\mathcal{F}_i)\) for the case where \(n \geq 2k_r + k_2 - t\) and \(\emptyset \neq \mathcal{F}_i \subseteq \{A \subseteq [n] \colon |A| \leq k_i\}\) for each \(i \in [r]\). By taking \(k_1 = \cdots = k_r = k\), \(\mu_1(k) = \dots = \mu_r(k) = 1\) and \(\mu_1(j) = \dots = \mu_r(j) = 0\) for each \(j \in \{0, 1, \dots, n\} \backslash \{k\}\), the maximum sum of sizes of \(r\) non-empty \(r\)-cross \(t\)-intersecting families of \(k\)-element subsets of \([n]\) is obtained.