This paper proposes a novel multi-task learning approach based on spatial-temporal recurrent imputation network (SRIN) for power system short-term voltage stability (STVS) assessment with incomplete PMU measurements. The state-of-the-art data imputation methods are based on single and separated learning tasks, which lack optimality for fully exploiting the information in available data. They are also facing several challenges in practical applications, e.g., dependence on complete datasets for training, and performance degradation under continuous data missing scenarios. As a significant advantage, the proposed SRIN method jointly optimizes the objective of missing value imputation and stability prediction through a multi-task recurrent network model. In this way, the integrated model can fully learn from any available data in the incomplete historical database, and the performance of both tasks can benefit from knowledge sharing and transferring across tasks. Moreover, the proposed method has superior advantages in handling both spatial and temporal consecutive missing scenarios, where the imputations are derived by an intelligent combination of history-based and feature-based estimations. Numerical simulation results on two test systems show that, under any PMU missing condition, the proposed method can maintain a competitively high STVS assessment accuracy with a much less imputation error. Note to Practitioners-This paper addresses the challenge of incomplete system observations for power system real-time stability assessment. This problem is not unique to power systems but also extends to other sequential prediction problems facing severe data incompleteness. Existing approaches to solve the missing data problem either relay on complete historical data to train an imputation model, which may not always hold true during practical applications, or impute the missing data by simple statistics, which lacks optimality and adaptivity under diverse missing patterns. This paper proposed a novel, integrated approach to solve this problem by jointly optimizing the two tasks together through a new recurrent network model. In this way, the method can fully learn from seriously undermined datasets. Moreover, this method deals with consecutive missing in time and space, by the design of a trainable weighting component. Numerical simulation results on standard power systems shows that the proposed multi-task model improve the performance of both two tasks and have high adaptivity to different data missing scenarios. In the future research, we will try to address the learning efficiency of this approach for application to larger systems and exploring its adaptability in more extreme scenarios.