  The Berry phase is a fundamental concept in quantum mechanics with profound
implications for understanding topological properties of quantum systems. This
tutorial provides a comprehensive introduction to the Berry phase, beginning
with the essential mathematical framework required to grasp its significance.
We explore the intrinsic link between the emergence of a non-trivial Berry
phase and the presence of topological characteristics in quantum systems,
showing the connection between the Berry phase and the band structure as well
as the phase's gauge-invariant nature during cyclic evolutions. The tutorial
delves into various topological effects arising from the Berry phase, such as
the quantum, anomalous, and spin Hall effects, which exemplify how these
quantum phases manifest in observable phenomena. We then extend our discussion
to cover the transport properties of topological insulators, elucidating their
unique behaviour rooted in Berry phase physics. This tutorial aims at equipping
its readers with a robust understanding of the basic theory around the Berry
phase and its pivotal role in the realm of topological quantum phenomena.
