Recently, deep learning is widely used in the field of remaining useful life (RUL) prediction. Among various deep learning technologies, recurrent neural network (RNN) and its variant, e.g., long short-term memory (LSTM) network, are gaining more attention because of their capability of capturing temporal dependence. Although the existing RNN-based approaches have demonstrated their RUL prediction effectiveness, they still suffer from the following two limitations: 1) it is difficult for RNN to extract directly degradation features from original monitoring data, and 2) most of the RNN-based prognostics methods are unable to quantify the uncertainty of prediction results. To address the above limitations, this paper proposes a new method named Residual convolution LSTM (RC-LSTM) network. In RC-LSTM, a new ResNet-based convolution LSTM (Res-ConvLSTM) layer is stacked with convolution LSTM (ConvLSTM) layer to extract degradation representations from monitoring data. Then, predicated on the RUL following a normal distribution, an appropriate output layer is constructed to quantify the uncertainty of the forecast result. Finally, the effectiveness and superiority of RC-LSTM is verified using monitoring data from accelerated degradation tests of rolling element bearings.