The rapid development of generative artificial intelligence (GenAI) technology has triggered extensive discussions about its potential applications in sustainable higher education. Based on the technology acceptance model (TAM) and task–technology fit (TTF) theory, this research aimed to investigate the current situations and challenges of Chinese university students using GenAI in four typical task scenarios. This was performed using a cross-sectional research design. The data were collected via questionnaire, with 486 undergraduates from a Chinese university participating. The data analysis methods include descriptive statistics, inferential statistics, and content analysis. The results show that more than 70% of university students actively use GenAI, but nearly half of them are not very proficient in its use. Doubao and ERNIE Bot are the GenAI tools they prefer most. The primary functions they use are text production and information retrieval. They mainly learn the relevant knowledge and skills through self-media and knowledge-sharing platforms. Among the four typical task scenarios, GenAI is widely used in course learning and research activities, while its application in daily life and job search is relatively limited. The analysis of demographic variables shows that grade and major have a significant impact on university students’ use of GenAI. In addition, university students suggest that universities should offer relevant courses or lectures and provide comprehensive technical support to improve the popularity and operability of GenAI. This study provides suggestions for universities, education administration departments, and technology development departments to improve GenAI services. It will help universities optimize the allocation of educational resources and promote educational equity for sustainability.