In this thesis,we study the existence of fully nontrivial ground-state solu-tions and synchronized ground-state solutions of the following coupled nonlinear Schrodinger system（?）for m=2,3 under suitable assumptions,where Ω（?）RN is a bounded domain or RN,N≥ 3,F（t1,t2…,tm）∈C1（Rm,R）,κi∈R,λ>0 is large enough and gi∈C（R）,（i=1,2,…,m）.This thesis is divided into five chapters,the main contents are organized as follows:In Chapter 1 we mainly introduced the physical background,discussed some results,gave suitable assumptions for this work and presented the main theorems.In Chapter 2 we provided some preliminary knowledge,then introduced two main propositions and a variational setting.Chapter 3 is devoted to the subcritical case of（0.1）.We consider the following system（?）where Ω（?）RN is a bounded domain,N≥ 3,F（t,s）∈ C1（R2,R）,κ1,κ2∈R,λ>0 and g1,g2 ∈ C（R）.Firstly,we make some assumptions for the functions g1,g2,then by establishing the Palias-Smale condition and Nehari manifold.Finally,Theorem 1.1 has been proved in this chapter.Chapter 4 deals with the critical case.In this chapter we study the systems（?）and（?）where μi>0,pij>1,pij+pji=2*,i≠j,i,j=1,2,3 and Ω is a bounded domain.By the minimization problem,we investigated the existence of a grounded-state solution which is synchronized for the former one.Also by variational method,we found a ground-state solution to（0.4）which is fully nontrivial.Finally,in Chapter 5 we revisit to the critical system（0.4）with κ1=κ2=κ3=κ to discuss ground-state solutions.By considering a minimizing problem as[26],we show that all ground state solutions are synchronized for 0<λ<λ1 and N≥4,where 0<λ1<λ2<…<λm<…are the eigenvalues of-Δ in H01（Ω）.