These lectures will cover a number of topics related to the role of symmetry principles in physics. After introducing some basic definitions and distinctions, we will look at the changing role of the relativity principle from Galileo to Einstein (in special and general relativity). In particular we raise doubts as to the wisdom of treating Einstein\'s 1905 route to special relativity based on the relativity principle as a template for a fundamental reformulation of quantum theory, as some have recently proposed. We then examine the nature and meaning of Noether\'s 1918 theorems for global and local symmetries, and discuss some recent applications of Noether\'s \"first\" theorem in quantum mechanics and electromagnetism. We finish with two case studies: (i) the Galilean (boost) symmetry of quantum mechanics, and (ii) the problem of the arrow of time in statistical mechanics based on time-reversal invariant dynamics. Each lecture will be followed by a period for discussion.