This paper presents a comprehensive review on the development of the modelling of ash deposition with particle combustion, sticking, rebound and removal behaviors. The modelling of ash deposit morphomology is also included. Ash deposition in coal and biomass fired boilers will induce many ash-related issues (such as slagging, fouling and corrosion) which will reduce the boiler efficiency and capacity. Some traditional prediction methods have been proposed to evaluate ash deposition. However, these methods are based on chemical compositions of ash deposits and the operating temperatures in boilers, which are unable to fully predict the complex ash deposition process. Great efforts have been made to develop mechanistic models to predict ash deposition processes in nature. The behavior of ash formation and deposition in the boilers plays a key role in the design of boilers and the selection of fuels. The ash formation process is primarily due to the fragmentation and coalescence of mineral matters in fuels and only a small portion of ash is formed. When ash particles impact the heat transfer surfaces, only a few particles will deposit on these surfaces and several ash deposition mechanisms have been identified to predict their behaviors, such as inertial impaction (for large particles), thermophoresis (for fine particles) and condensation (for vapors). The ash deposition mechanisms used in the experimental, numerical and mechanistic studies coupled with the fuels and investigated systems are summarized in this paper.Numerous attempts have been made to develop different models to overcome the shortcomings of traditional methods. As such, various numerical attempts have been made to predict the growth behavior of ash deposition in furnaces and boilers by employing comprehensive combustion models coupled with high fidelity computational fluid dynamics (CFD) modelling methods for different types of fuels. Furthermore, several combustion codes have been incorporated into the ash deposition models, including the fuel combustion process (the release of volatiles, devolatilization and char combustion), wall reaction and consumption sub-models as well as the packed bed and overbed combustion sub-models. Moreover, several ash deposition sub-models have been developed to predict the ash deposition growth behavior using computational fluid dynamics (CFD) methods in combustors of different scales. In order to better understand the impact behavior of ash particles, some analytical models such as dynamic models and kinetic models have been developed. For accurate prediction of impaction efficiency, an impaction correction factor has been proposed to reduce the effect of coarse meshes on the impaction efficiency. Also, the stickiness of ash particles which is determined by kinetic energy, viscosity and molten degree of ash particles plays a key role in the ash formation and deposition processes and determines whether ash particles stick on the surfaces or rebound from the surfaces. Briefly, three main types of particle adhesion theories are used to evaluate if an impacting particle bounces off or sticks to the surface, namely, the viscosity-based empirical model, the critical velocity model and the melt fraction model.When the particles impact the heat transfer surfaces, they may rebound from the surface or remove the ash deposit. The particle surface energy with its static contact angle is also important in determining the sticking efficiency. In addition, several rebound criteria have been proposed to predict the particle rebound behavior, which include the critical rebound velocity, energy balance, excess energy, bouncing potential and critical impact angle on a flat or oblique plate or on a heat transfer tube. Tremendous efforts have also been made to develop the theories and mechanisms of ash deposition as well as removal on the surfaces of heat exchangers, some of which were based on the Kern-Seaton theory. Furthermore, several removal sub-models have been proposed to predict the particle removal behavior, including the energy balance, moment conservation, energy dissipation, critical moment theory, critical shear velocity and critical impact angle. In the actual fouling process, the growth behavior of fouling on the tube surfaces changes the original tube shape continuously. The fly ash deposited on the surface alters the boundary of heat transfer and affects the distribution of the temperature, flow field as well as the deposition rates. Various theoretical methods have been proposed to predict the ash deposit morphology, including the lattice Boltzmann method (LBM), dynamic meshing method, and time and mass magnification factors. In addition, many investigations have focused on developing models for inter-particle thermal conductivity by studying the ash deposit microstructure to characterize the thermal and morphological changes through an ash deposit. Several ash deposition layer models have been experimentally and theorectically studied in details, including the two-layer, three-layer, four-layer and six-layer sub-models.