Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. When some eigenvalues of $X_0$ separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of $X_0$. These findings are largely due to Benaych-Georges and Rochet, Bordenave and Capitaine, and Tao. When all eigenvalues of $X_0$ lie inside the unit disk, we prove that local eigenvalue statistics at the spectral edge form a new class of determinantal point processes. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. This talk is based on joint work with Lu Zhang (张禄,中科大), arXiv:2204.13171v2.