Existing automatic prompt engineering methods are typically designed for discriminative tasks, where new task prompts are iteratively refined with limited feedback from a single metric reflecting a single aspect. However, these approaches are suboptimal for generative tasks, which require more nuanced guidance beyond a single numeric metric to improve the prompt and optimize multiple aspects of the generated text. To address these challenges, we propose a novel multi-aspect Critique-Suggestion-guided automatic Prompt Optimization (CriSPO) approach. CriSPO introduces a critique-suggestion module as its core component. This module spontaneously discovers aspects, and compares generated and reference texts across these aspects, providing specific suggestions for prompt modification. These clear critiques and actionable suggestions guide a receptive optimizer module to make more substantial changes, exploring a broader and more effective search space. To further improve CriSPO with multi-metric optimization, we introduce an Automatic Suffix Tuning (AST) extension to enhance the performance of task prompts across multiple metrics. We evaluate CriSPO on 4 state-of-the-art LLMs across 4 summarization and 5 QA datasets. Extensive experiments show 3-4% ROUGE score improvement on summarization and substantial improvement of various metrics on QA. Code available at https://github.com/amazon-science/crispo