  Multivariate orthogonal polynomials in $D$ real dimensions are considered
from the perspective of the Cholesky factorization of a moment matrix. The
approach allows for the construction of corresponding multivariate orthogonal
polynomials, associated second kind functions, Jacobi type matrices and
associated three term relations and also Christoffel-Darboux formul{\ae}. The
multivariate orthogonal polynomials, its second kind functions and the
corresponding Christoffel-Darboux kernels are shown to be quasi-determinants
--as well as Schur complements-- of bordered truncations of the moment matrix;
quasi-tau functions are introduced. It is proven that the second kind functions
are multivariate Cauchy transforms of the multivariate orthogonal polynomials.
Discrete and continuous deformations of the measure lead to Toda type
integrable hierarchy, being the corresponding flows described through Lax and
Zakharov-Shabat equations; bilinear equations are found. Varying size matrix
nonlinear partial difference and differential equations of the 2D Toda lattice
type are shown to be solved by matrix coefficients of the multivariate
orthogonal polynomials. The discrete flows lead to expressions for the
multivariate orthogonal polynomials and its second kind functions in terms of
shifted quasi-tau matrices, which generalize to the multidimensional realm
those that relate the Baker and adjoint Baker functions with ratios of Miwa
shifted $\tau$-functions in the 1D scenario. In this context, the multivariate
extension of the elementary Darboux transformation is given in terms of
quasi-determinants of matrices built up by the evaluation, at a poised set of
nodes lying in an appropriate hyperplane in $\mathbb R^D$, of the multivariate
orthogonal polynomials. The multivariate Christoffel formula for the iteration
of $m$ elementary Darboux transformations is given as a quasi-determinant.
