The authors extend the notion of the quadratic numerical range of a \(2\times 2\) block operator matrix recently introduced in [J. Oper. Theory 39, 339–359 (1998; Zbl 0996.47006)]. Let \(n\in{\mathbb N}\), \(H_1,\dots,H_n\) be complex Hilbert spaces, \({\mathcal H}=H_1\times \dots \times H_n\), \(i,j=1,\dots, n\), and let \({\mathcal A}=(A_{ij})_{i,j=1}^{n}\in L({\mathcal H})\), where \(A_{ij}\in L(H_j, H_i)\), \(i,j=1,\dots, n\). For \(x=(x_1,\dots,x_n)\in{\mathcal H}\) define \({\mathcal A}_x\in M_n({\mathbb C})\) (the space of \(n\times n\) matrices over \({\mathbb C}\)) by \({\mathcal A}_x= ((A_{ij}x_j, x_i))^{n}_{i,j=1}\). The block numerical range \(W_{h_1\times\dots\times H_n}({\mathcal A})=W^n({\mathcal A})\) of an \(n\times n\) block operator matrix \({\mathcal A}\) is defined as the set of all \(\lambda\in{\mathbb C}\) for which there exist \(x_1\in H_1,\dots\), \(x_n\in H_n\), \(\| x_1\| =\dots=\| x_n\| =1\), such that \(\det ({\mathcal A}_x -\lambda I_n)=0\), where \(I_n\) denotes the identity matrix in \({\mathbb C^n}\). The main results of this paper concern spectral inclusion, inclusion between block numerical ranges for refined block decompositions, an estimate of the resolvent in terms of the block numerical range, and block numerical ranges of companion operators.