The Batalin-Vilkovisky (BV) formalism plays a fundamental role in the
quantization of gauge theories, yet its rigorous formulation within the
framework of noncommutative geometry (NCG) remains an open challenge.
In this talk, I will present recent results solving this problem for
U(n)-gauge theories arising from finite spectral triples. A key aspect
of the BV construction is the extension of the initial gauge theory
through the introduction of ghost fields, which I will formalize using
the new notion of BV-spectral triple. I will then discuss how the
BV/BRST cohomology complexes naturally align within the NCG framework,
establishing connections with appropriate Hochschild complexes.