D-brane gauge theories from toric singularities of the form $C^3\mathrm{/}\mathrm{\Gamma }$ and $C^4\mathrm{/}\mathrm{\Gamma }$

We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifolds ${\mathbf{C}}^{\mathbf{3}}\mathbf{/}{\mathbf{Z}}_{\mathbf{2}}\times {\mathbf{Z}}_{\mathbf{2}}$,${\bf C^3/Z_2\times Z_3}$and${\bf C^4/Z_2\times Z_2 \times Z_2}$.