This paper presents a comprehensive survey of the relaxed-type equations, a class of partial differential equations that extend the Fourier and Fickian-type equations by incorporating the finite propagation speed concept or micro-structural interactions influences. It explores the theoretical foundations of such equations, examines their mathematical properties, and highlights their significance in modeling different physical phenomena in a variety of applications. Moreover, recent advances and future prospects from various scientific fields are provided, opening the door for further research in this dynamic and developing field of study.