Banded linear systems arise in many communication scenarios, e.g., those involving inter-carrier interference and inter-symbol interference. Motivated by recent advances in deep learning, we propose to design a high-accuracy low-complexity signal detector for banded linear systems based on convolutional neural networks (CNNs). We develop a novel CNN-based detector by utilizing the banded structure of the channel matrix. Specifically, the proposed CNN-based detector consists of three modules: the input preprocessing module, the CNN module, and the output postprocessing module. With such an architecture, the proposed CNN-based detector is adaptive to different system sizes, and can overcome the curse of dimensionality, which is a ubiquitous challenge in deep learning. Through extensive numerical experiments, we demonstrate that the proposed CNN-based detector outperforms conventional deep neural networks and existing model-based detectors in both accuracy and computational time. Moreover, we show that CNN is flexible for systems with large sizes or wide bands. We also show that the proposed CNN-based detector can be easily extended to near-banded systems such as doubly selective orthogonal frequency division multiplexing (OFDM) systems and 2-D magnetic recording (TDMR) systems, in which the channel matrices do not have a strictly banded structure.