  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.
