Advanced wave engineering demands robust, efficient, and new-degree-of-freedom manipulations in wave intrinsic properties like amplitude, phase, polarization, and even spatial structure. To tackle this, this thesis advances photonic wave manipulation by systematically exploiting topological effects for wave steering, trapping, and emission phenomena in scattering systems and engineered metamaterials. The focused photonic topological effects are categorized into standard band topology leading to Floquet topological insulators (anomalous and Chern phase) and emergent topological defects contributing to various scattering singularities (scattering zero, bound state in the continuum, coherent perfect absorption, exceptional point), giving rise to enhanced control over wave dynamics in spatial, temporal, and momentum domains.

The main achieved results are organized into three interconnected and progressive thrusts.

Topological wave steering: Spatially, electrically tunable topological domain wall efficiently and rapidly reconfigures the unidirectional edge states in heterostructured Floquet insulators with superior robustness. Temporally, scalar magnetic hysteresis is enriched with nonreciprocal scattering and induces topological hysteretic winding and phase transitions around defects like coherent perfect absorption and exceptional point, showing improved singularity-topology interplay for anti-lasing and non-Hermitian braiding.

Topological wave trapping: The bound state in the continuum (BIC), a lossless resonance that remains localized even though it exists within free space, represents a critical advancement in wave trapping by defying conventional wisdom. A scattering version of BIC is demonstrated by creating a real-space disclination state in a nonreciprocal topological insulator. The momentum-space topological nature of BIC is explored and experimentally achieved towards the fully disordered case and three-dimensional photonic crystal, pushing the topological wave trapping towards real-world applications in wavefront engineering and vortex generation.

Topological wave emitting: A simple symmetric cube with holes hosting three-dimensional near-field singularities is integrated with gain medium to create a multidimensional vortex maser, resulting in the first observation of microwave vortex-like photon emission from each surface of the cubic cavity.

These results culminate in functional topological devices across microwave and optical platforms such as reconfigurable waveguides, topological emitters, disorder-and-imperfection-immune photonic networks and free-space meta-devices. By unifying concepts from topological insulators, topological defects, and non-Hermitian physics, this work expands the toolkit for controlling wave-matter interactions, offering pathways to robust photonic technologies, optical communication, structured light, ultrasensitive sensing, and temporal control. It suggests the potential for topological active metasurfaces, higher-order topologies, and compact topological light sources in the near future.