The rapid advancements and stringent requirements of modern embedded computing systems have led to a surge in the demand for customized processors that can efficiently cater to specific application needs. This survey paper delves into the realm of automating application-driven customization of extensible processors, offering insights into the challenges, advancements, and trends in this domain. It explores the trade-offs between fine-grained and coarse-grained customization, discussing Custom Instructions (CIs) identification and optimization techniques, while emphasizing the shift towards larger accelerators that target complex control sequences in the application. It scrutinizes the balance between speedup and reusability, addressing the challenges of efficient design approaches to manage area and power consumption. The agile nature of early Design Space Exploration (DSE) is discussed, where rapid evaluation of area and communication costs plays a pivotal role. In essence, this survey serves as a valuable guide for researchers and practitioners in the field of processor customization, aiding designers in navigating this complex landscape to optimize performance in a rapidly evolving computing paradigm.