Let \(\phi(x,t): \mathbb{R}^N\times [0,\infty)\to [0,\infty)\) be a convex function of \(x\), satisfying the \(\Delta_2\)-condition for all \(t\geq 0\), defining a Musiełak-Orlicz space \(L^\phi(G)\), \(G\) being an open set in \(\mathbb{R}^N\). The authors define the \((k,\Phi)\)-capacity of \(E\) relative to \(G\), where \(E\subset\mathbb{R}^N\), by the formula \[ C_{k,\phi}(E, G)= \text{inf}\Biggl\{\int_G \phi(y,f(y))\,dy;\;f\in S_k(E,G)\Biggr\}, \] where \(k\) is a kernel function on \(\mathbb{R}^N\) and \(S_k(E,G)\) is the family of all nonnegative measurable functions on \(\mathbb{R}^N\) vanishing outside \(G\), such that \(k*f(x)\geq 1\) for every \(x\in E\). There are studied basic properties of such capacities and investigated their estimates on balls.