Nontrivial topology in physical systems is the driving force behind many interesting phenomena. Notably, phases of matter must be classified in part by their topological properties. Phases with topological (nonlocal) order, such as the fractional quantum Hall effect (QHE), can support anyonic excitations obeying fractional statistics with potential application in topological quantum computing. States lacking intrinsic topological order can still belong to topological phases, provided certain symmetries are imposed. On their own, these symmetry-protected phases do not support anyons, but they can still have other interesting features, such as protected boundary states. In this thesis we present an original research into several topologically nontrivial systems. First, we present the experimental and theoretical results on light propagation in the valley modes of inversion-symmetry broken honeycomb lattices (HCL). We find that a rotating spiral pattern, leading to Zitterbewegung, arises in the intensity profile of the beam as a result of the nontrivial topology of the HCL valleys. Next, we present the numerical demonstration of dynamical topological phase transitions driven by nonlinearity of the photonic medium, which occur in soliton SSH lattices. The phase transitions, marked by the appearance of topological edge states in the band gap, occur due to the setup which enables continually changing relative values of the intracell and intercell soliton couplings. Finally, we propose a scheme for creating and manipulating synthetic anyons in a noninteracting system by perturbing it with specially tailored localized probes. The external probes are needed because noninteracting systems do not possess the kind of topological order required to support anyonic excitations. We start from a noninteracting 2D electron gas in a uniform magnetic field, which is in an integer QHE state, and introduce thin solenoids carrying a fractional magnetic flux. We present the solution for a suitable ground state, and demonstrate the fractional braiding statistics in the coordinates of the solenoids.