  We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian
$Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their
Poincar\'e-Einstein metrics $(X,g_+)$. Our approach is based on the spectral
theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor
valued meromorphic family of distributions with residues given by the residue
family operators $\slashed{D}_N^{res}(h;\lambda)$ on spinors. We develop basic
aspects and properties of $\slashed{D}_N^{res}(h;\lambda)$ including conformal
covariance, factorization properties by conformally covariant operators for
both flat and curved semi-Riemannian $Spin$-manifolds, and Poisson
transformation.
