  The deformed $W$-algebra is a quantum deformation of the $W$-algebra ${\cal
W}_\beta(\mathfrak{g})$ in conformal field theory. Using the free field
construction, we obtain a closed set of quadratic relations of the $W$-currents
of the deformed $W$-algebra. This allows us to define the deformed $W$-algebra
by generators and relations. In this review, we study two types of deformed
$W$-algebra. One is the deformed $W$-algebra ${\cal
W}_{x,r}\big(A_{2N}^{(2)}\big)$, and the other is the $q$-deformed corner
vertex algebra $q$-$Y_{L_1, L_2, L_3}$ that is a generalization of the deformed
$W$-algebra ${\cal W}_{x,r}\big(A(M,N)^{(1)}\big)$ via the quantum toroidal
algebra.
