This paper is a largely expository account of the theory of p x p matrix polyno mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n … , hn of p x p matrices with h-i = hj for j = 0, … , n. We let k = O, … ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, … , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, … ,n, (1.2) k II} . ‘-0 ’ I- whenever Hk is invertible.