In this paper, we define the grand mixed Morrey generalized spaces \(\mathcal{M}^{P),\Psi (.)}_{u}(\mathbb{R}^{n})\) over Euclidean spaces, where \(P=(p_{1} ,\ldots,p_{n})\) , \(\Psi (.)=(\psi_{1} (.) ,\ldots,\psi_{n} (.))\) is an \(n\) -tuple of positive increasing functions \(\psi_{i}\) defined on \((0,p_{i}-1]\) with \(p_{i}\in(1,\infty)\) and \(i= 1 ,\ldots,n\) , and \(u(\cdot,\cdot)\) is a positive Lebesgue measurable function defined on \(\mathbb{R}^{n}\times(0,\infty)\) . We establish embedding and density properties for spaces \(\mathcal{M}^{P),\Psi (.),\Sigma}_{u}(\mathbb{R}^{n})\) with \(0<\Sigma<P-1\) . As applications, we prove that the bilinear \(\omega\) -type Calderón-Zygmund operator \(\widetilde{T}_{\omega}\) and its commutator \(\widetilde{T}_{\omega,b_{1},b_{2}}\) which is formed by \(b_{1}, b_{2}\in\mathrm{BMO}(\mathbb{R}^{n})\) and \(\widetilde{T}_{\omega}\) are bounded from the product of grand mixed generalized Morrey spaces \(\mathcal{M}^{P_{1}),\Theta,\Sigma_{1}}_{u_{1}}(\mathbb{R}^{n})\times \mathcal{M}^{P_{2}),\Theta,\Sigma_{2}}_{u_{2}}(\mathbb{R}^{n})\) into the spaces \(\mathcal{M}^{P),\Theta,\Sigma}_{u}(\mathbb{R}^{n})\) , and they are also bounded from the product of grand mixed Morrey spaces \(M^{P_{1}),\Theta,\Sigma_{1}}_{q_{1}}( \mathbb{R}^{n})\times M^{P_{2}),\Theta,\Sigma_{2}}_{q_{2}}(\mathbb{R}^{n})\) into the spaces \(M^{P),\Theta,\Sigma}_{q}(\mathbb{R}^{n})\) , where \(u_{1}u_{2}=u\) , \(\Theta=(\theta_{1} ,\ldots,\theta_{n})>0\) , \(\frac{1}{P}= \frac{1}{P_{1}} +\frac{1}{P_{2}}\) for \(1<P_{1}, P_{2}<\infty\) , \(0<\Sigma_{i}=(\sigma_{i1},\sigma_{i2} ,\ldots,\sigma_{in})<P_{i}-1\) \((i=1,2)\) and \(\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}\) for \(1< q_{1}, q_{2}<\infty\) .
