  The purpose of this work is to classify, for given integers $m,\, n\geq 1$,
the bordism class of a closed smooth $m$-manifold $X$ with a free smooth
involution $\tau$ with respect to the validity of the {\it Borsuk-Ulam
property} that for every continuous map $\phi : X \to R^n$ there exists a point
$x\in X$ such that $\phi (x)=\phi (\tau (x))$. We will classify a given free
$Z_2$-bordism class $\alpha$ according to the three possible cases that (a) all
representatives $(X , \tau)$ of $\alpha$ satisfy the Borsuk-Ulam property; \
(b) there are representatives $(X_ 1, \tau_1)$ and $(X_2, \tau_2)$ of $\alpha$
such that $(X_1, \tau_1)$ satisfies the Borsuk-Ulam property but $(X_2,
\tau_2)$ does not; \ (c) no representative $(X , \tau)$ of $\alpha$ satisfies
the Borsuk-Ulam property.
