The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p,1}^{n/p-1}\left({ℝ}^n\right)\times {\dot{B}}_{p,1}^{n/p}\left({ℝ}^n\right)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.