In this talk, we define the notion of structured pseudospectra. We
prove
that for the structures Toeplitz and circulant, the structured
pseudospectrum equals the unstructured pseudospectrum. We show that this
is false for Hermitian and skew-Hermitian structures. We generalize the
result to pseudospectra of matrix polynomials. Indeed, we prove that the
structured pseudospectrum equals the unstructured pseudospectrum for
matrix polynomials with Toeplitz, circulant and Hankel structures. We
conclude by giving a formula for structured pseudospectra of real matrix
polynomials.