We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let K_{1},\ldots,K_{n} be Borel subsets of \mathbb{R}^{m_{1}},\ldots,\mathbb{R}^{m_{n}} respectively, and \pi :\mathbb{R}^{m_{1}} \times \cdots \times \mathbb{R}^{m_{n}}\rightarrow \mathbb{R}^{k} be a surjective linear map. We set \mathfrak{m}: = \mathrm{\min }⁡\Bigg\{\sum \limits_{i \in I}\mathrm{\dim }_{H}⁡(K_{i}) + \mathrm{\dim }⁡\pi \Bigg(\bigoplus \limits_{i \in I^{c}}\mathbb{R}^{m_{i}}\Bigg),\:I \subset \{1,\ldots,n\},\:I \neq \emptyset \Bigg\}. Consider the space \Lambda _{m} = \{(t,O),\:t \in \mathbb{R},\:O \in \mathrm{SO}(m)\} with the natural measure and set \Lambda = \Lambda _{m_{1}} \times\cdots\times \Lambda _{m_{n}} . For every \lambda = (t_{1},O_{1},\ldots,t_{n},O_{n}) \in \Lambda and every x = (x^{1},…,x^{n}) \in \mathbb{R}^{m_{1}} \times \cdots \times \mathbb{R}^{m_{n}} we define \pi _{\lambda }(x) = \pi (t_{1}O_{1}x^{1},\ldots,t_{n}O_{n}x^{n}) . Then we have Theorem. (i) If \mathfrak{m} > k , then \pi _{\lambda }(K_{1} \times \cdots \times K_{n}) has positive k-dimensional Lebesgue measure for almost every \lambda \in \Lambda . (ii) If \mathfrak{m}⩽k and \mathrm{\dim }_{H}⁡(K_{1} \times\cdots \times K_{n}) = \mathrm{\dim }_{H}⁡(K_{1}) + \cdots + \mathrm{\dim }_{H}⁡(K_{n}) , then \mathrm{\dim }_{H}⁡(\pi _{\lambda }(K_{1} \times … \times K_{n})) = \mathfrak{m} for almost every \lambda \in \Lambda .