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We study differential-difference equation of the form $$\frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x)) $$ with unknown$t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equationof such kind is called Darboux integrable, if there exist two functions $F$ and$I$ of a finite number of arguments $x$, $\{t(n\pm k,x)\}_{k=-\infty}^\infty$,${\frac{d^k}{dx^k}t(n,x)}_{k=1}^\infty$, such that $D_xF=0$ and $DI=I$, where$D_x$ is the operator of total differentiation with respect to $x$, and $D$ isthe shift operator: $Dp(n)=p(n+1)$. Reformulation of Darboux integrability interms of finiteness of two characteristic Lie algebras gives an effective toolfor classification of integrable equations. The complete list of Darbouxintegrable equations is given in the case when the function $f$ is of thespecial form $f(u,v,w)=w+g(u,v)$.

Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$