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In this paper, we study the spreading speeds and traveling wave solutions ofthe PDE $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) +u(1-u),\ \ x\in\mathbb{R}^N 0=\Delta v-v+u, \ \ x\in\mathbb{R}^N, \end{cases}$$ where $u(x,t)$ and $v(x,t)$ represent the population and the chemoattractantdensities, respectively, and $\chi$ is the chemotaxis sensitivity. It has beenshown in an earlier work by the authors of the current paper that, when$0<\chi<1$, for every nonnegative uniformly continuous and bounded function$u_0(x)$, the system has a unique globally bounded classical solution$(u(x,t;u_0),v(x,t;u_0))$ with initial condition $u(x,0;u_0)=u_0(x)$.Furthermore, if $0<\chi<\frac{1}{2}$, then the constant steady-state solution$(1,1)$ is asymptotically stable with respect to strictly positiveperturbations. In the current paper, we show that if $0<\chi<1$, then there arenonnegative constants $c_{-}^*(\chi)\leq c_+^*(\chi)$ such that for everynonnegative initial function $u_0(\cdot)$ with nonempty compact support $$\lim_{t\to\infty} \sup_{|x|\le ct} \big[|u(x,t;u_0)-1|+|v(x,t;u_0)-1|\big]=0\ \\forall\ 0<c<c_{-}^*(\chi) $$ and $$ \lim_{t\to\infty}\sup_{|x|\geq ct} \big[u(x,t;u_0)+v(x,t;u_0)\big]=0\ \ \forall\ c>c_{+}^*(\chi).$$ We also show thatif $0<\chi<\frac{1}{2}$, there is a positive constant $c^*(\chi)$ such that forevery $c\ge c^*(\chi)$, the system has a traveling wave solution$(u(x,t),v(x,t))$ with speed $c$ and connecting $(1,1)$ and $(0,0)$, that is,$(u(x,t),v(x,t))=(U(x-ct),V(x-ct))$ for some functions $U(\cdot)$ and$V(\cdot)$ satisfying $(U(-\infty),V(-\infty))=(1,1)$ and$(U(\infty),V(\infty))=(0,0)$. Moreover, $$ \lim_{\chi\to0}c^*(\chi)=\lim_{\chi\to 0}c_+^*(\chi)=\lim_{\chi\to 0}c_-^*(\chi)=2. $$ Wefirst give a detailed study in the case $N=1$ and next we extend these resultsto the case $N\ge 2$.

Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N