Periodic driving has recently been investigated as a mechanism for generating nontrivialtopological phases of matter within otherwise ordinary systems. Periodic driving caneven induce new, so-called anomalous topological phases, which have no counterpart inequilibrium. This thesis studies such topological phenomena and phases in periodicallydriven systems. The first part of the thesis introduces the concept of topological phasesin periodically driven systems, and classifies the noninteracting topological phasesthat can arise in such systems, including the anomalous phases. The second part ofthe thesis studies the anomalous Floquet insulator (AFI), which is an example of ananomalous topological phase. The discussion here shows that the AFI is characterizedby a quantized, nonzero bulk magnetization density, and demonstrates that strongdisorder can stabilize the phase in the presence of interactions. The third part ofthe thesis explores driving-induced topological effects in other physical systems. Thediscussion here shows that periodic driving can lead to new, topologically-robustenergy pumping effects. In some cases, these effects can be described as fully classicalphenomena and have potentially useful applications. A novel master equation fordissipative, periodically driven quantum systems is derived in this connection