We consider the problem of computing the determinant of a matrix of polynomials. Four algorithms are compared: expansion by minors, Gaussian elimination over the integers, a method based on evaluation and interpolation, and a procedure which computes the characteristic polynomial of the matrix. Each method is analyzed with respect to its computing time and storage requirements using several models for polynomial growth. The results show which method is preferable for a given computational model. In addition to these asymptotic results, the analysis is exactly done for certain especially small, yet practical and important cases. Key Words: determinants, matrix of polynomials, Gaussian elimination, expansion by minors, characteristic polynomial.