During the last twenty years mathematics and physics have significantly influenced each other and became highly entangled. Mathematical physics was always producing a wide variety of new concepts and problems that became important subjects of the pure mathematical research. The growth of gauge, gravity and string theories have made the relation between these subjects closer than ever before. An important driving force was the discovery of quantum groups and of the gauge/gravity dualities. Here the leading role was played by the the so-called AdS/CFT duality and the underlying integrable structure of it. A far-reaching concept is the effect of boundaries and the corresponding boundary conditions. They are unavoidable in almost all models of mathematical physics and are of the fundamental importance. The introduction of boundaries into the theory of quantum groups leads to a whole new class of the so-called reflection algebras. Such algebras were shown to appear in numerous mathematical models and are at the core of the integrable structure of them. Furthermore, these algebras were also shown to play a prominent role in the AdS/CFT. However a coherent framework for describing such algebras is not known, and many properties of the reflection algebras are still an open question. The goal of this research is to develop new algebraic methods and intradisciplinary connections between the axiomatic theory of algebras and the theory of quantum groups inspired by the integrable structure of the AdS/CFT, in particular by shedding more light on the effects of boundaries and different boundary configurations. The research is driven by applying algebraic objects such as the quantum affine and Yangian algebras to find elegant, exact solutions describing the models that arise from and are inspired by the gauge/gravity dualities.