  This article, based on the author's PhD thesis, reviews recent advancements
in the field of quantum integrability, in particular the separation of
variables (SoV) program for high-rank integrable spin chains and the boost
mechanism for solving the Yang-Baxter equation. We begin with a general
overview of quantum integrable systems with special emphasis on their
description in terms of quantum algebras. We then provide a detailed account of
the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and
T- and Q-systems. We then introduce the notion of separation of variables in
integrable systems and build on Sklyanin's work in rank 1 models and extend to
higher rank. By exploiting a novel link between SoV and quantum algebra
representation theory we construct the separated variables for
$\mathfrak{gl}(n)$ spin chains for arbitrary compact representations of the
symmetry algebra and develop various new tools along the way. Next, we build on
the previous part and develop a new technique for the computation of scalar
products in the SoV framework which we call Functional SoV or FSoV. Unlike the
work in the previous, operatorial, part this approach is based on the Baxter TQ
equations. After developing this technique we supplement it with a new operator
construction providing a unified view of functional and operatorial SoV. Then,
we generalise the results of the previous part from compact spin chains to
non-compact spin chains. The final part of this work is based on the
development of tools for solving the Yang- Baxter equation. We develop a
bottom-up approach for this based on the so-called Boost automorphism and uses
the spin chain Hamiltonian as a starting point. Our approach allows us to
classify numerous families of solutions in particular a complete classification
of 4 x 4 solutions which preserve fermion number which have applications in the
AdS/CFT correspondence.
