In this survey paper, the authors discuss Higgs bundles over complex manifolds, and focus on Bruzzo's conjecture which characterizes semistable Higgs bundles with vanishing discriminant in terms of their behaviour when restricted to curves. In Section~2, the \(\mu\)-stability and the Gieseker stability for vector bundles over a polarized variety \((X, H)\) are reviewed. The properties of \(\mu\)-semistable bundles, including Bogomolov's inequality, are presented. Section~3 begins with the definition of Higgs bundles. A Higgs bundle on a variety \(X\) is a pair \((E, \phi)\), where \(E\) is a vector bundle on \(X\) and \(\phi : E \to E \otimes \Omega_X^1\) is a morphism satisfying \(\phi \wedge \phi = 0\) (\(\phi\) is called the Higgs field). A variety \(X\) is defined to be a Higgs variety if for any Higgs bundle \((E, \phi)\) on \(X\), \((E, \phi)\) is \(\mu\)-semistable with vanishing discriminant if and only if for any morphism \(f : C \to X\), where \(C\) is a smooth irreducible projective curve, the Higgs bundle \(f^*(E, \phi)\) is \(\mu\)-semistable. A conjecture due to U.~Bruzzo asserts that every smooth projective variety is a Higgs variety. Three equivalent statements for Bruzzo's Conjecture are analysed. In Section~4, the authors present an overview of all partial results (toward Bruzzo's Conjecture) obtained to date, and give examples of Higgs varieties. In Section~5, the relation with Tannakian categories is explained.