In the present paper, a general subclass ${\mathcal{M}}_{{\Sigma}_m}^{h,p}(\lambda,\gamma)$ of the $m$-Fold symmetric bi-univalent functions is defined. Also, the estimates of the Taylor-Maclaurin coefficients $|a_{m+1}|$, $|a_{2m+1}|$ and Fekete-Szegö problems are obtained for functions in this new subclass. The results presented in this paper would generalize and improve some recent works of several earlier authors.