In this paper we mainly focus on some determinants with Legendre symbol entries. For an odd prime $p$ and an integer $d$, let $S(d,p)$ denote the determinant of the $(p-1)/2\times(p-1)/2$ matrix whose $(i,j)$-entry $(i,j=1,…,(p-1)/2)$ is the Legendre symbol $((i^2+dj^2)/p)$. We investigate properties of $S(d,p)$ as well as some other determinants. In Section 3 we pose 17 conjectures on determinants one of which states that $\big(\frac{-S(d,p)}p\big)=1$ if $(\frac dp)=1$. This material might interest some readers and stimulate further research. ; Comment: 19 pages. Expand some remarks to reflect certain progress