Dynamics of outwardly-propagating reaction fronts inspires both fundamental and practical interests. While for the former we can mention in pass supernovae explosions and measurements of laminar flame speeds of reactive mixtures, the latter is relevant to the operation of engines and prevention of spark-initiated hazards such as accidental or intentional explosions. While a pointy-ignited, expanding premixed flame front is initially smooth, it is subsequently gets corrugated by diffusional-thermal (cellular or pulsating) instability. Eventually, with the flame thickness being reduced as compared to the global flame radius, the onset of hydrodynamic instability initiates the continuous production of cascades of cells over the flame surface, which increases the total flame surface area as compared to the respective spherical flame. Hence, the flame accelerates. The present study is devoted to the latter, self-accelerative stage of expanding flame propagation. Based on the logarithmic transformation of Akkerman et al. [PRE 83 (2011) 026305], we develop a self-similar formulation on accelerative propagation of wrinkled flames, describing the relevant dynamic and scalar fields, namely, the velocity and temperature profiles in the fresh medium. We next consider trajectories of fresh gas particles and analyze the possibility of detonation initiation through the heating transfer ahead of the accelerating flame front by the compression waves generated ahead of it. The flame dynamics depends on whether combustion occurs in free/confined, laminar /turbulent environment. While we start with laminar flames in the free space, the analysis is subsequently extended to the turbulent environment. It is reported that an expanding flame front in externally-forced, near-isotropic turbulence exhibit accelerative propagation given by a well-defined power law, with much larger acceleration exponent as compared to that in the laminar environment. The formulation is also extended to confined flames, with a model for flame-acoustic coupling incorporated in the analysis. Specifically, the sound waves modify the power-law flame acceleration based on the average global flame radius, facilitating or inhibiting the transition to detonation, which allows a reconciliation of a discrepancy in experimental measurements of different groups. Finally, we also consider expanding flames in central gravitational field. The independent and competitive roles of hydrodynamic and body-force instabilities are analyzed, and the state of termination of a globally-spherical flame front is estimated. It is demonstrated that while intrinsic combustion instabilities are presumably too weak to trigger the detonation in terrestrial conditions, this effect should nevertheless be accounted along with other phenomena. In contrast, enormous acceleration of expanding flames, with subsequent detonation, can occur at the astrophysical scales.