The paradigm of b-geometry (Melrose) is useful for studying differential operators on a manifold M that suffer a first order degeneracy along a hypersurface Z. The b-tangent bundle is defined to have as sections the vector fields on M that are tangent along Z. Substituting the b-tangent bundle for the usual tangent bundle, one obtains b-geometry analogues for various classical geometries (symplectic b-geometry, Riemannian b-geometry, etc). Mendoza introduced complex b-geometry. In this talk, I will describe recent joint work with Tatyana Barron establishing the Newlander-Nirenberg theorem for complex b-manifolds (arXiv:2310.08013). Time permitting, I will connect this work to previous and ongoing work of mine on singular foliations.