  The aim of this memoir for "Habilitation \`a Diriger des Recherches" is to
present quantum geometric and algebraic aspects of supersymmetric gauge theory,
which emerge from non-perturbative nature of the vacuum structure induced by
instantons. We start with a brief summary of the equivariant localization of
the instanton moduli space, and show how to obtain the instanton partition
function and its generalization to quiver gauge theory and supergroup gauge
theory in three ways: the equivariant index formula, the contour integral
formula, and the combinatorial formula. We then explore the geometric
description of $\mathcal{N} = 2$ gauge theory based on Seiberg-Witten geometry
together with its string/M-theory perspective. Through its relation to
integrable systems, we show how to quantize such a geometric structure via the
$\Omega$-deformation of gauge theory. We also discuss the underlying quantum
algebraic structure arising from the supersymmetric vacua. We introduce the
notion of quiver W-algebra constructed through double quantization of
Seiberg-Witten geometry, and show its specific features: affine quiver
W-algebras, fractional quiver W-algebras, and their elliptic deformations.
