This is a survey article on the modeling of thermoelastic waves in a solid body. The term ''Generalized Thermoelasticity'' stands for a Hyperbolic Thermoelasticity in which a thermomechanical load applied to a body is transmitted in a wave-like manner throughout the body. Only transient thermoelastic waves are included in the survey. The modeling covers: (i) L-S model (Lord and Shulman, 1967), (ii) G-L model (Green and Lindsay, 1972), (iii) H-I model (Hetnarski and Ignaczak, 1996), (iv) G-N model (Green and Naghdi, 1993), and (v) C-T model (Chandrasekharaiah and Tzou, 1998). Attention focuses on the theoretical and practical significance of these models. Theoretical results include the domain of influence theorems, uniqueness as well as one-dimensional closed-form solutions. Except for a limiting case when the models reduce to rigid heat conductors, none of the five models has been verified by a combined thermoelastic experiment up to date, so their applications to real thermoelastic bodies present a challenge for future generations of researchers in the field.