An expanding spherical flame is hydrodynamically unstable in the flame-sheet limit, attained either as the flame reaches a sufficiently large dimension as compared to the flame thickness, and/or when it propagates in a high-pressure environment such that its thickness is correspondingly reduced. The cells that continuously develop over the flame surface increase its area and thereby the global propagation rate, resulting in the possibility of self-acceleration. The present study examines whether this self-acceleration could be self-similar, and if so whether it could also be self-turbulizing. A critical appraisal of the experimental and computational results in the literature on these issues was performed, and experiments were conducted for hydrogen/air mixtures over an extensive range of elevated pressures. Results demonstrate the strong possibility of self-similar flame acceleration, moderate influences of diffusionalthermal instability and of the system pressure, and a corresponding moderate spread in the power-law acceleration exponent.