Laguerre-Gaussian (LG) laser beams are called vortex beams because of their optical vortices, points of zero intensity centered on the beam axis which the beam of light twists around like a corkscrew. Topological charge (TC) is a quantity that describes the number of windings in an LG beam's helical phase front, and is equal to the index l in the LG beam's e^il azimuthal phase term. These beams have generated significant interest recently following the realization that each photon in them carries a quantity l of orbital angular momentum (OAM). Vortex beams have immense potential for application in nuclear physics: JLab is considering generating vortex gamma ray sources, focusing vortex beams on photocathodes for polarized electron sources, and creating high power vortex beams with Fabry-P rot interferometers to probe nuclear structure. These new tools may shed light on fundamental problems like the proton spin crisis. We investigate the propagation of vortex beams diffracted by circular and triangular apertures theoretically and experimentally. The properties of these diffracted beams are of interest as the diffraction patterns depend on the LG beam's TC sign and magnitude, and can thus be used to measure their OAM directly. Simulations indicate that the diffraction patterns reach a stable state in the far field, where they become symmetrical and vortices propagate without change, and we find good agreement in our experimental results. We also briefly discuss the possibility of performing Compton backscattering with Gaussian beams that carry OAM.