The talk will give an overview of some recent research trends involving operator algebras and dynamics, with an emphasis on work in which I have myself been involved, together with coauthors Larry Baggett, Ola Bratteli, Berndt Brenken, Dorin Dutkay, Kathy Merrill, Judy Packer, Steen Pedersen, Myung-Sin Song; among others. The use of dynamics center around recursive iteration schemes as they are used for example in Iterated Function Systems (IFSs), a special class of fractals, in wavelet algorithms, and in such applied areas as signal and image processing. A leading operator algebraic theme will be a certain class of representations of the Cuntz algebras.