Abstract. We study the asymptotic behaviour of the eigenvalues of Hermitian n × n block Toeplitz matrices An,m, with m × m Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function f , and we study their eigenvalues for large n and m, relating their behaviour to some properties of f as a function; in particular we show that, for any fixed k, the first k eigenvalues of An,m tend to inf f , while the last k tend to sup f , so extending to the block case a well-known result due to Szegö. In the case the An,m's are positive-definite, we study the asymptotic spectrum of P −1 n,m An,m, where Pn,m is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system An,mx = b, obtaining strict estimates, when n and m are fixed, and exact limit values, when n and m tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.