The effects of channel curvature on the flame propagation are examined for premixed flames in micro channels. Previous studies have shown that curved channels can enhance the flame stretching, which can significantly increase the flame propagation speed. In this study, the effect of flame stretching was excluded by the narrow channel limit in order to study the effects of channel curvature on the flame propagation. An asymptotic approach was first extended to include the effects of channel curvature. Accordingly, the two-dimensional problem was reduced to an effective one-dimensional problem with a clear physical meaning of curvature. A parametric study of curvature was then performed to investigate its effects on the flame propagation. It was found that curvature inhibits both the convection and diffusion processes. An increase in the curvature results in a significantly reduced flame thickness and a significantly increased reaction rate. Furthermore, the initial flame propagation speed is smaller in the channel with a larger curvature, but its growth rate is higher under the same ignition condition. With a large channel curvature, the flame has a smaller propagation speed of the steady compressibility-driven flame and a shorter distance to reach the steady state.