In this paper the Ⅰ and Ⅱ regular n-simplices are introduced. We prove that the sufficient and necessary conditions for existence of an Ⅰ regular n-simplex in Rn are that if n is even then n= 4m(m + 1), and if n is odd then n= 4m + 1 with that n + 1 can be expressed as a sum of two integral squares or n = 4m - 1, and that the sufficient and necessary condition for existence of a Ⅱ regular n-simplex in Rn is n= 2m2 - 1 or n= 4m(m+1)(m ( ) N). The connection between regular n-simplex in Rn and combinational design is given.