The background field method is presented for multi-loop calculations in gauge field theories. It allows for the computation of a gauge-invariant effective action without losing explicit gauge invariance. The method is equivalent to that of 't Hooft, but follows more closely the conventional functional approach. The gauge field is split into a background field A and a quantum field Q. The background field gauge ensures gauge invariance in terms of A. The effective action is shown to be equal to the conventional effective action evaluated in an unusual gauge. Renormalization is discussed, and it is shown that the renormalization programme can be carried out without reference to fields inside loops. The renormalization of the gauge coupling constant, background field, and gauge-fixing parameter is required. The coupling constant and background field renormalizations can be determined from the A field two-point function. The method allows for the calculation of the β function for pure Yang-Mills theory. The one- and two-loop contributions to the β function are calculated using the background field approach. The β function is determined by calculating the background field two-point function. The method simplifies calculations by avoiding the need for vertex functions. The two-loop calculation shows that the β function is given by β = -g(β₀(g²/4π²) + β₁(g⁴/4π⁴)), where β₀ = (11/3)C_A and β₁ = (34/3)C_A². The calculation confirms the results of previous studies. The background field method provides a gauge-invariant effective action and simplifies renormalization procedures. The method is particularly useful for gauge theory calculations. The results demonstrate the effectiveness of the background field approach in maintaining gauge invariance and simplifying renormalization.The background field method is presented for multi-loop calculations in gauge field theories. It allows for the computation of a gauge-invariant effective action without losing explicit gauge invariance. The method is equivalent to that of 't Hooft, but follows more closely the conventional functional approach. The gauge field is split into a background field A and a quantum field Q. The background field gauge ensures gauge invariance in terms of A. The effective action is shown to be equal to the conventional effective action evaluated in an unusual gauge. Renormalization is discussed, and it is shown that the renormalization programme can be carried out without reference to fields inside loops. The renormalization of the gauge coupling constant, background field, and gauge-fixing parameter is required. The coupling constant and background field renormalizations can be determined from the A field two-point function. The method allows for the calculation of the β function for pure Yang-Mills theory. The one- and two-loop contributions to the β function are calculated using the background field approach. The β function is determined by calculating the background field two-point function. The method simplifies calculations by avoiding the need for vertex functions. The two-loop calculation shows that the β function is given by β = -g(β₀(g²/4π²) + β₁(g⁴/4π⁴)), where β₀ = (11/3)C_A and β₁ = (34/3)C_A². The calculation confirms the results of previous studies. The background field method provides a gauge-invariant effective action and simplifies renormalization procedures. The method is particularly useful for gauge theory calculations. The results demonstrate the effectiveness of the background field approach in maintaining gauge invariance and simplifying renormalization.