Speech separation can be formulated as learning to estimate a time-frequency mask from acoustic features extracted from noisy speech. For supervised speech separation, generalization to unseen noises and unseen speakers is a critical issue. Although deep neural networks (DNNs) have been successful in noise-independent speech separation, DNNs are limited in modeling a large number of speakers. To improve speaker generalization, a separation model based on long short-term memory (LSTM) is proposed, which naturally accounts for temporal dynamics of speech. Systematic evaluation shows that the proposed model substantially outperforms a DNN-based model on unseen speakers and unseen noises in terms of objective speech intelligibility. Analyzing LSTM internal representations reveals that LSTM captures long-term speech contexts. It is also found that the LSTM model is more advantageous for low-latency speech separation and it, without future frames, performs better than the DNN model with future frames. The proposed model represents an effective approach for speaker- and noise-independent speech separation.