In this talk, we will introduce some recent findings about the smallest singular value and the bottom singular vector of a heavy-tailed rectangular random matrix with entry tail exponent 0 < alpha < 4. In the regime 2 < alpha < 4, we will show a Tracy–Widom to Gaussian phase transition for the smallest singular value when alpha crosses 8/3. In the regime 0 < alpha < 2, we will show some partial results on upper and lower bounds for the smallest singular value. Finally, we will show a localization–delocalization transition for the bottom singular vector when alpha crosses 2. This talk is based on joint work with Jaehun Lee and Xiaocong Xu.