  In this paper we study Hardy and Rellich type inequalities for
Baouendi-Grushin vector fields : $\nabla_{\gamma}=(\nabla_x,
|x|^{2\gamma}\nabla_y)$ where $\gamma>0$, $\nabla_x$ and $\nabla_y$ are usual
gradient operators in the variables $x\in \mathbb{R}^m$ and $y\in\mathbb{R}^k$,
respectively. In the first part of the paper, we prove some weighted Hardy type
inequalities with remainder terms. In the second part, we prove two versions of
weighted Rellich type inequality on the whole space. We find sharp constants
for these inequalities. We also obtain their improved versions for bounded
domains.
