The abstract: In this talk I give a brief review of our recent development on the cosmological dynamics in the Einstein-Gauss-Bonnet gravity. The special attention is paid to the models which allow realistic compactification. We require only for model to be free of future singularities and has a ``standard'' cosmological singularity af the origin. Using such weak requirements, for the simplest spatially flat model, we are able to put constraints on the parameters of the theory and the number of extra dimensions. Consideration of more complex
models (with spatial curvature and additional anisotropies) allow us to tighten the constraints.
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