In this paper, the existence of finite traveling waves (FTWs) for the reaction-diffusion system 1 $$\frac{{\partial U_1 }}{{\partial t}} = \frac{{\partial ^2 U_1 }}{{\partial x^2 }} + b\frac{{\partial U_1 }}{{\partial x}} - \mathop {{\text{II}}}\limits_{J = 1}^N U_J^m 1J,x \in R,t > 0,i = 1,2,..,N,N \geqslant 3$$ (I) is studied, where b is a real number, 1 $$ m_{iJ} \geqslant 0(i,j = 1,2,...,N){\text{ and }}\sum\nolimits_{j = {\text{1}}}^N {m_{ij} > 0,(i = 1,2,...,N)} $$ . A solution 1 $$U(x,t) = (U_1 (x,t),U_2 (x,t),...,U_N (x,t)){\text{ of (I) of the form }}U_i (x,t) = u_i (x + ct),c \in R,i = 1,2,...,N$$ with the properties that there exists a finite;γo ε Rsuch that 1 $$u,(y) = 0{\text{ for }}y \leqslant y_0 ,u_1 (y) \in C^2 (R),{\text{ and }}u_1 (y)$$ is nonnegative, nondecreasing and nontrivial, is called an FTW of (I). It is proved that if all principal minors of I — A are positive, where $$A = (m_{ij} )_{N{\text{ x }}N} $$ then (I) has a unique FTW for any c ε R. Otherwise (I) has no FTWs.