  In this paper we define the fundamental quandle of knotoids and linkoids and
prove that it is invariant under the under forbidden-move and hence encodes
only the information of the underclosure of the knotoid. We then introduce
$n$-pointed quandles, which generalize quandles by specifying $n$ elements as
ordered basepoints. This leads to the notion of fundamental pointed quandles of
linkoids, which enhances the fundamental quandle. Using $2$-pointed quandle
allows us to distinguish 1-linkoids with equivalent under-closures and leads to
a couple of 1-linkoid invariants. We also generalize the notion of homogeneity
of quandles to $n$-homogeneity of quandles. We classify all
$\infty$-homogeneous, finite quandles.
