FROM COMPACT QUANTUM GROUPS TO MULTIPLIER HOPF ALGEBRAS: the Peter-Weyl theory of compact quantum groups (existence of the Haar measure, dense *-Hopf subalgebras, representation theory), universal compact quantum groups and free products of compact quantum groups (Banica and Wang), Rieffel's deformations (quantizations) of Lie groups, C*-algebraic quantum groups from multiplier Hopf algebras