For a locally compact abelian group $G$, Fuglede's conjecture states that a Borel set is spectral if and only if it tiles the group $G$ by translation. In the case $G=\mathbb{R}^n$, it have been studied for long time since Fuglede formulated this conjecture in 1974. It is proved to be false for $n\ge 3$ but it is still open for $n=1,2$. With A.Fan, S.Fan and L.Liao, we consider the case $G=\mathbb{Q}_p$ the field of p-adic numbers and give an affirmative answer to the conjecture in this case. Moreover, we prove that the spectral sets in $\mathbb{Q}_p$ are compact open up to a null set. With A.Fan and S.Fan, we give a geometric criterion of spectral compact open sets, which is called $p$-homogeneity. In this talk, I will first give an overview of Fuglede's conjecture and then talk about $p$-homogeneity.