Functionally graded materials (FGMs) are a new class of advanced engineering materials being developed and implemented in industry, which eliminate the material property discontinuities by grading material composition and continuously varying the compositions or volume fractions. Due to rapidly growing research interests in these materials, ways to monitor their performance and gather information on their properties are also becoming a focus for researchers. In this study, a guided wave-based non-destructive evaluation technique for the testing and monitoring of damage in functionally graded beams is developed. Firstly, functionally graded materials and the types of damage that can occur in them are reviewed and identified. Then the characteristics of guided waves propagating in functionally graded materials (FGMs) is determined analytically and numerically to optimize their application in a structural health monitoring (SHM) and non-destructive evaluation (NDE) system for the detection of damage. In numerical modelling and simulation, a finite element model is devised to simulate guided wave propagating in functionally graded beams made of functionally graded carbon fiber-phenolic nanocomposites (FGCN). The numerical results offer insights into the complicated and often erratic nature of guided waves that propagate in functionally graded materials as well as their interaction with damage.