I begin these lectures by examining the transformation properties of quantum fields under the discrete symmetries of Parity, P, Charge Conjugation, C, and Time Reversal, T. With these results in hand, I then show how the structure of the Standard Model helps explain the conservation/violation of these symmetries in various sectors of the theory. This discussion is also used to give a qualitative proof of the CPT Theorem, and some of the stringent tests of this theorem in the neutral Kaon sector are reviewed. In the second part of these lectures, global symmetries are examined. Here, after the distinction between Wigner-Weyl and Nambu-Goldstone realizations of these symmetries is explained, a discussion is given of the various, approximate or real, global symmetries of the Standard Model. Particular attention is paid to the role that chiral anomalies play in altering the classical symmetry patterns of the Standard Model. To understand the differences between anomaly effects in QCD and those in the electroweak theory, a discussion of the nature of the vacuum structure of gauge theories is presented. This naturally raises the issue of the strong CP problem, and I present a brief discussion of the chiral solution to this problem and of its ramifications for astrophysics and cosmology. I also touch briefly on possible constraints on, and prospects for, having real Nambu-Goldstone bosons in nature, concentrating specifically on the simplest example of Majorons. I end these lectures by discussing the compatibility of having global symmetry in the presence of gravitational interactions. Although these interactions, in general, produces small corrections, they can alter significantly the Nambu-Goldstone sector of theories.

50 pages, 2 figures, Latex file