1D Convolutional Neural Networks (CNNs) have recently become the state-of-the-art technique for crucial signal processing applications such as patient-specific ECG classification, structural health monitoring, anomaly detection in power electronics circuitry and motor-fault detection. This is an expected outcome as there are numerous advantages of using an adaptive and compact 1D CNN instead of a conventional (2D) deep counterparts. First of all, compact 1D CNNs can be efficiently trained with a limited dataset of 1D signals while the 2D deep CNNs, besides requiring 1D to 2D data transformation, usually need datasets with massive size, e.g., in the "Big Data" scale in order to prevent the well-known "overfitting" problem. 1D CNNs can directly be applied to the raw signal (e.g., current, voltage, vibration, etc.) without requiring any pre- or post-processing such as feature extraction, selection, dimension reduction, denoising, etc. Furthermore, due to the simple and compact configuration of such adaptive 1D CNNs that perform only linear 1D convolutions (scalar multiplications and additions), a real-time and low-cost hardware implementation is feasible. This paper reviews the major signal processing applications of compact 1D CNNs with a brief theoretical background. We will present their state-of-the-art performances and conclude with focusing on some major properties.