  We consider the problem of reconstruction of a non-linear finite-parametric
model $M=M_p(x),$ with $p=(p_1,...,p_r)$ a set of parameters, from a set of
measurements $\mu_j(M)$. In this paper $\mu_j(M)$ are always the moments
$m_j(M)=\int x^jM_p(x)dx$. This problem is a central one in Signal Processing,
Statistics, and in many other applications.
  We concentrate on a direct (and somewhat "naive") approach to the above
problem: we simply substitute the model function $M_p(x)$ into the measurements
$\mu_j$ and compute explicitly the resulting "symbolic" expressions of
$\mu_j(M_p)$ in terms of the parameters $p$. Equating these "symbolic"
expressions to the actual measurement results, we produce a system of nonlinear
equations on the parameters $p$, which we consequently try to solve.
  The aim of this paper is to review some recent results (mostly of
\cite{Vet5,Mil1,Mil2,Put1,Vet4,Vet3,Mil3,Mil4,Vet2}) in this direction,
stressing the algebraic structure of the arising systems and mathematical tools
required for their solutions.
  In particular, we discuss the relation of the reconstruction problem above
with the recent results of \cite{bfy,bry,chr,pak1,pak2,pak3,pry,ry} on the
vanishing problem of generalized polynomial moments and on the Cauchy-type
integrals of algebraic functions.
  The accompanying paper \cite{Kis1} (this volume) provides a solution method
for a wide class of reconstruction problems as above, based on the study of
linear differential equations with rational coefficient, which are satisfied by
the moment generating function of the problem.
