  The concept of Gauss quadrature can be generalized to approximate linear
functionals with complex moments. Following the existing literature, this
survey will revisit such generalization. It is well known that the (classical)
Gauss quadrature for positive definite linear functionals is connected with
orthogonal polynomials, and with the (Hermitian) Lanczos algorithm.
Analogously, the Gauss quadrature for linear functionals is connected with
formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm
with look-ahead strategy; moreover, it is related to the minimal partial
realization problem. We will review these connections pointing out the
relationships between several results established independently in related
contexts. Original proofs of the Mismatch Theorem and of the Matching Moment
Property are given by using the properties of formal orthogonal polynomials and
the Gauss quadrature for linear functionals.
