Classifying multivariate time series (MTS), which record the values of multiple variables over a continuous period of time, has gained a lot of attention. However, existing techniques suffer from two major issues. First, the long-range dependencies of the time-series sequences are not well captured. Second, the interactions of multiple variables are generally not represented in features. To address these aforementioned issues, we propose a novel Cross Attention Stabilized Fully Convolutional Neural Network (CA-SFCN) to classify MTS data. First, we introduce a temporal attention mechanism to extract long- and short-term memories across all time steps. Second, variable attention is designed to select relevant variables at each time step. CA-SFCN is compared with 16 approaches using 14 different MTS datasets. The extensive experimental results show that the CA-SFCN outperforms state-of-the-art classification methods, and the cross attention mechanism achieves better performance than other attention mechanisms.