Traditional speaker diarization seeks to detect ``who spoke when'' according to speaker characteristics. Extending to target speech diarization, we detect ``when target event occurs'' according to the semantic characteristics of speech. We propose a novel Multimodal Target Speech Diarization (MM-TSD) framework, which accommodates diverse and multi-modal prompts to specify target events in a flexible and user-friendly manner, including semantic language description, pre-enrolled speech, pre-registered face image, and audio-language logical prompts. We further propose a voice-face aligner module to project human voice and face representation into a shared space. We develop a multi-modal dataset based on VoxCeleb2 for MM-TSD training and evaluation. Additionally, we conduct comparative analysis and ablation studies for each category of prompts to validate the efficacy of each component in the proposed framework. Furthermore, our framework demonstrates versatility in performing various signal processing tasks, including speaker diarization and overlap speech detection, using task-specific prompts. MM-TSD achieves robust and comparable performance as a unified system compared to specialized models. Moreover, MM-TSD shows capability to handle complex conversations for real-world dataset.