This survey paper examines recent advancements in low-resolution signal processing, emphasizing quantized compressed sensing. Rising costs and power demands of high-sampling-rate data acquisition drive the interest in quantized signal processing, particularly in wireless communication systems and Internet of Things sensor networks, as 6G aims to integrate sensing and communication within cost-effective hardware. Motivated by this urgency, this paper covers novel signal processing algorithms designed to address practical challenges arising from quantization and modulo operations, as well as their impact on system performance. We begin by introducing the framework of one-bit compressed sensing and discuss relevant theories and algorithms. We explore the application of quantized compressed sensing algorithms to sensor networks, radar, cognitive radio, and wireless channel estimation. We highlight how generic methods can be tailored to an application using specific examples from wireless channel estimation. Additionally, we review other low-resolution techniques beyond one-bit compressed sensing along with their applications. We also provide a brief overview of the emerging concept of unlimited sampling. While this paper does not aim to be exhaustive, it selectively highlights results to inspire readers to appreciate the diverse algorithmic tools (convex optimization, greedy methods, and Bayesian approaches) and sampling techniques (task-based quantization and unlimited sampling).