This paper presents exact and approximate methods for computing the distribution of quadratic forms in normal variables. The focus is on calculating the probability $ P\{Q > x\} $, where Q is a quadratic form. The paper discusses the decomposition of quadratic forms into sums of independent chi-squared variables, which is essential for computing their distributions. For non-central quadratic forms, the distribution is expressed as a combination of chi-squared variables with non-centrality parameters. The paper also provides a finite expression for the probability $ P\{Q > x\} $, derived from the characteristic function and involving derivatives of a function related to the distribution. For non-central quadratic forms, the probability density function is derived using Bessel functions and involves integrating the inversion formula of the characteristic function. The paper then presents a numerical method for inverting the characteristic function, which allows for the computation of the cumulative distribution function. This method involves integrating an expression derived from the characteristic function and using numerical techniques to approximate the integral. The accuracy of the method is discussed, and it is shown that the trapezoidal rule provides more accurate results than Simpson's rule for certain cases. The paper also compares different approximations for the distribution of quadratic forms, including Pearson's three-moment chi-squared approximation and Patnaik's two-moment chi-squared approximation. It is shown that the three-moment approximation is more accurate, particularly in the upper tail of the distribution. The paper concludes with a discussion of the computational methods used, including the use of the Ferranti 'Mercury' computer at CERN for many of the computations. The results are summarized in Table 1, which provides probabilities for various quadratic forms and compares them with different approximations.This paper presents exact and approximate methods for computing the distribution of quadratic forms in normal variables. The focus is on calculating the probability $ P\{Q > x\} $, where Q is a quadratic form. The paper discusses the decomposition of quadratic forms into sums of independent chi-squared variables, which is essential for computing their distributions. For non-central quadratic forms, the distribution is expressed as a combination of chi-squared variables with non-centrality parameters. The paper also provides a finite expression for the probability $ P\{Q > x\} $, derived from the characteristic function and involving derivatives of a function related to the distribution. For non-central quadratic forms, the probability density function is derived using Bessel functions and involves integrating the inversion formula of the characteristic function. The paper then presents a numerical method for inverting the characteristic function, which allows for the computation of the cumulative distribution function. This method involves integrating an expression derived from the characteristic function and using numerical techniques to approximate the integral. The accuracy of the method is discussed, and it is shown that the trapezoidal rule provides more accurate results than Simpson's rule for certain cases. The paper also compares different approximations for the distribution of quadratic forms, including Pearson's three-moment chi-squared approximation and Patnaik's two-moment chi-squared approximation. It is shown that the three-moment approximation is more accurate, particularly in the upper tail of the distribution. The paper concludes with a discussion of the computational methods used, including the use of the Ferranti 'Mercury' computer at CERN for many of the computations. The results are summarized in Table 1, which provides probabilities for various quadratic forms and compares them with different approximations.