Intersection of quadrics in ${mathbb{C}}^n$, moment-angle manifolds, complex manifolds and convex polytopes

These are notes for the CIME school on Complex non-Kahler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-Kahler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations in an open set in complex projective space ${{mathbb P}_{mathbb C}}^{n-1}$