  The full moduli space M of a class of N=1 supersymmetric gauge theories is
studied. For gauge theories living on a stack of D3-branes at Calabi-Yau
singularities X, M is a combination of the mesonic and baryonic branches, the
former being the symmetric product of X. In consonance with the mathematical
literature, the single brane moduli space is called the master space F.
Illustrating with a host of explicit examples, we exhibit many
algebro-geometric properties of the master space such as when F is toric
Calabi-Yau, behaviour of its Hilbert series, its irreducible components and its
symmetries. In conjunction with the plethystic programme, we investigate the
counting of BPS gauge invariants, baryonic and mesonic, using the geometry of F
and show how its refined Hilbert series not only engenders the generating
functions for the counting but also beautifully encode ``hidden'' global
symmetries of the gauge theory which manifest themselves as symmetries of the
complete moduli space M for arbitrary number of branes.
