Score-based Generative Models (SGMs) have demonstrated remarkable generalization capabilities, \eg generating unseen, but natural data. However, the greater the generalization power, the more likely the unintended generalization, and the more dangerous the abuse. Despite these concerns, research on unlearning SGMs has not been explored. To fill this gap, we first examine the current `gold standard' in Machine Unlearning (MU), \ie, re-training the model after removing the undesirable training data, and find it does not work in SGMs. Further analysis of score functions reveals that the MU ‘gold standard’ does not alter the original score function, which explains its ineffectiveness. Building on this insight, we propose the first Moderated Score-based Generative Model (MSGM), which introduces a novel score adjustment strategy that redirects the score function away from undesirable data during the continuous-time stochastic differential equation process. Albeit designed for SGMs, MSGM is a general and flexible MU framework compatible with diverse diffusion architectures, training strategies and downstream tasks. The code will be shared upon acceptance.