Global conformal invariants are integrals of geometric scalars which remain invariant under conformal changes of the underlying metric. I will discuss (parts of) my recent proof of a conjecture of Deser and Schwimmer, which states that any such global invariant can be decomposed into standard "building blocks" of three types. (This lecture will be introductory and self-contained; it will be followed by three other technical lectures in the Wedensday seminars where I will present more ideas from the proof).