  This is a review article on the quantum toroidal algebras, focusing on their
roles in various solvable structures of 2d conformal field theory,
supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as
our starting point, we elucidate the interconnection of affine Yangians,
quantum toroidal algebras, and double affine Hecke algebras.
  Our exploration delves into the representation theory of the quantum toroidal
algebra of $\mathfrak{gl}_1$ in full detail, highlighting its connections to
partitions, $\mathcal{W}$-algebras, Macdonald functions, and the notion of
intertwiners. Further, we also discuss integrable models constructed on Fock
spaces and associated $\mathcal{R}$-matrices, both for the affine Yangian and
the quantum toroidal algebra of $\mathfrak{gl}_1$.
  The article then demonstrates how quantum toroidal algebras serve as a
unifying algebraic framework that bridges different areas in physics. Notably,
we cover topological string theory and supersymmetric gauge theories with eight
supercharges, incorporating the AGT duality. Drawing upon the representation
theory of the quantum toroidal algebra of $\mathfrak{gl}_1$, we provide a
rather detailed review of its role in the algebraic formulations of topological
vertex and $qq$-characters. Additionally, we briefly touch upon the corner
vertex operator algebras and quiver quantum toroidal algebras.
