  This paper offers a review of numerical methods for computation of the
eigenvalues of Hermitian matrices and the singular values of general and some
classes of structured matrices. The focus is on the main principles behind the
methods that guarantee high accuracy even in the cases that are ill-conditioned
for the conventional methods. First, it is shown that a particular structure of
the errors in a finite precision implementation of an algorithm allows for a
much better measure of sensitivity and that computation with high accuracy is
possible despite a large classical condition number. Such structured errors
incurred by finite precision computation are in some algorithms e.g. entry-wise
or column-wise small, which is much better than the usually considered errors
that are in general small only when measured in the Frobenius matrix norm.
Specially tailored perturbation theory for such structured perturbations of
Hermitian matrices guarantees much better bounds for the relative errors in the
computed eigenvalues. % Secondly, we review an unconventional approach to
accurate computation of the singular values and eigenvalues of some notoriously
ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and
Hankel matrices. The distinctive feature of accurate algorithms is using the
intrinsic parameters that define such matrices to obtain a non-orthogonal
factorization, such as the \textsf{LDU} factorization, and then computing the
singular values of the product of thus computed factors. The state of the art
software is discussed as well.
