The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the special case of row and column matrices. The fully interlacing property is shown to be preserved under matrix products, flips across the reverse diagonal and Veronese sections of the power series involved. These results and their corollaries generalize, unify and simplify several results which have previously appeared in the literature. An application to the theory of uniform triangulations of simplicial complexes is included.