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It is known that the antipodal set of a Riemannian symmetric space of compact type $G/ K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 96 and 6, respectively; the Grassmanian $Gr_{r,r+q}(\mathbb{R})$ of type BD I has a $r(q-r)$-dimensional orbit as antipodal set, if $r\geq 5$ and $q>0$; and $SO(2r+1)$ has an antipodal set of dimension $2r$.

A complete description of the antipodal set of most symmetric spaces of compact type