  This paper provides a thorough exploration of the absolute value equations
$Ax-|x|=b$, a seemingly straightforward concept that has gained heightened
attention in recent years. It is an NP-hard and nondifferentiable problem and
equivalent with the standard linear complementarity problem. Offering a
comprehensive review of existing literature, the study delves into theorems
concerning the existence and nonexistence of solutions to the absolute value
equations, along with numerical methods for effectively addressing this complex
equation. Going beyond conventional approaches, the paper investigates
strategies for obtaining solutions with minimal norms, techniques for
correcting infeasible systems, and other pertinent topics. By pinpointing
challenging issues and emphasizing open problems, this paper serves as a
valuable guide for shaping the future research trajectory in this dynamic and
multifaceted field.
