Prompt-based tuning has been proven effective for pretrained language models (PLMs). While most of the existing work focuses on the monolingual prompts, we study the multilingual prompts for multilingual PLMs, especially in the zero-shot cross-lingual setting. To alleviate the effort of designing different prompts for multiple languages, we propose a novel model that uses a unified prompt for all languages, called UniPrompt. Different from the discrete prompts and soft prompts, the unified prompt is model-based and language-agnostic. Specifically, the unified prompt is initialized by a multilingual PLM to produce language-independent representation, after which is fused with the text input. During inference, the prompts can be pre-computed so that no extra computation cost is needed. To collocate with the unified prompt, we propose a new initialization method for the target label word to further improve the model's transferability across languages. Extensive experiments show that our proposed methods can significantly outperform the strong baselines across different languages. We will release data and code to facilitate future research.