Spin textures, such as non-uniform domain arrangements, domain walls and skyrmions are naturally occurring structures in magnetic materials. Recently, the unique properties of spin textures such as their reconfigurability, stability and scalability towards nanoscale dimensions, has sparkled the interests towards their use as active elements in spintronic devices. In the framework of magnonics, which aims to use spin waves for carrying and processing information, using spin textures allows to harness an extremely rich phenomenology for designing new functionalities. In this review, we focus on the recent developments on the control and stabilization of engineered spin textures, and their applications in the field of magnonics. First, we introduce the main techniques used for stabilizing the spin textures and controlling their properties. Then we review the recent results on the use of engineered spin textures for guiding, emitting and manipulating spin waves, and the recent proposals on the realization of complex functionalities using integrated spin-texture-based systems, which hint to possible future directions for technological prospects.