The process of the electromagnetic wave propagation in the 2D nonlinear Bragg grating structure has been studied. The cubic Kerr nonlinearity of materials was considered. The values of the cut-off frequencies were estimated with the modification of the effective refractive index method. The possibility of propagation of the signal which frequency lies within the band gap was demonstrated. The process of soliton tunneling and the multiple gap-soliton formation in the nonlinear 2D Bragg grating has been investigated for the different values of the structure parameters. Nonlinear dynamics in the tilted grating was also studied in detail.