Although phased arrays have been around for almost half a century, many digital signal processing techniques were not possible in practical systems until recently. These developments have been fuelled not only by new computational capabilities but also by the huge interest in phased arrays for civil applications such as massive multiple-input multiple-output (MIMO) communications, automotive radar and navigating unmanned aerial vehicles (UAVs). Motivated by the recent advances in hardware for antenna arrays and computational platforms, in this tutorial, we present a digital signal processing perspective of phased arrays as well as a gentle introduction to algorithms for direction-of-arrival (DoA) estimation and digital beamforming. Differently from hardware-oriented tutorials, in this tutorial we first establish a mathematical model for phased arrays from a signal processing perspective, providing the required background to introduce advanced array processing algorithms. Then, we present the foundations of classical beamforming and DoA estimation methods that have been successfully employed across different applications in the last years. Also, we will discuss modern techniques for array processing, going beyond classical subspace-based techniques, leveraging optimization theory and sparse signal processing for both regular and sparse phased arrays. There, not only the standard signal-based processing is considered but also more advanced covariance-based techniques, allowing for an increased number of degrees of freedom. In the last part of the tutorial, we delve into performance-based sparse array design, which fits into the field of sparse sensing. Here both convex and submodular optimization methods will be presented to design optimal sparse arrays for DoA estimation. Finally, highlights of other current developments in phased array signal processing algorithms and future research trends in communications and imaging systems will be presented.