  The formalism of Wiener filtering is developed here for the purpose of
reconstructing the large scale structure of the universe from noisy, sparse and
incomplete data. The method is based on a linear minimum variance solution,
given data and an assumed \prior model which specifies the covariance matrix of
the field to be reconstructed. While earlier applications of the Wiener filter
have focused on estimation, namely suppressing the noise in the measured
quantities, we extend the method here to perform both prediction and dynamical
reconstruction. The Wiener filter is used to predict the values of unmeasured
quantities, such as the density field in un-sampled regions of space, or to
deconvolve blurred data. The method is developed, within the context of linear
gravitational instability theory, to perform dynamical reconstruction of one
field which is dynamically related to some other observed field. This is the
case, for example, in the reconstruction of the real space galaxy distribution
from its redshift distribution When the field to be reconstructed is a Gaussian
random field, such as the primordial perturbation field predicted by the
canonical model of cosmology, the Wiener filter can be pushed to its fullest
potential. In such a case the Wiener estimator coincides with the Bayesian
estimator designed to maximize the {\it posterior} probability. The Wiener
filter can be also derived by assuming a quadratic regularization function, in
analogy with the `Maximum Entropy' method. The mean field obtained by the
minimal variance solution can be supplemented with constrained realizations of
the Gaussian field to
