This chapter on conservation laws and group theory in physics focuses primarily on Noether’s theorem, then on the unitary representations of the Galilean group, crucial for describing symmetries in both classical field theories and quantum mechanics. It begins by discussing Noether currents and the role of gauge choices on conserved quantities before addressing the challenge of constructing finite-dimensional unitary representations for non-compact groups like translations in infinite-dimensional spaces, then for compact groups to deal with rotations. The discussion highlights how differential operators represent translations and rotations, illustrating their action on scalar fields and their differential properties in the case of infinite-dimensional unitary representations. Boosts and time translations are similarly treated, emphasizing their differential operator representations. In quantum mechanics, these symmetries require unitary representations that account for phase factors due to the projective nature of quantum states. The chapter concludes with insights into the Bargmann algebra, which extends the Galilean algebra to describe non-relativistic quantum systems, and discusses its implications for physical observables like energy and momentum, linking theoretical group theory with practical applications in quantum mechanics.