Human action recognition in 3D skeleton sequences has attracted a lot of research attention. Recently, long short-term memory (LSTM) networks have shown promising performance in this task due to their strengths in modeling the dependencies and dynamics in sequential data. As not all skeletal joints are informative for action recognition, and the irrelevant joints often bring noise which can degrade the performance, we need to pay more attention to the informative ones. However, the original LSTM network does not have explicit attention ability. In this paper, we propose a new class of LSTM network, global context-aware attention LSTM, for skeleton-based action recognition, which is capable of selectively focusing on the informative joints in each frame by using a global context memory cell. To further improve the attention capability, we also introduce a recurrent attention mechanism, with which the attention performance of our network can be enhanced progressively. Besides, a two-stream framework, which leverages coarse-grained attention and fine-grained attention, is also introduced. The proposed method achieves state-of-the-art performance on five challenging datasets for skeleton-based action recognition.