  A set without tangents in $\PG(2,q)$ is a set of points S such that no line
meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set
of points $\E$ such that all secant lines of $\E$ are external lines of
$\mathcal{C}$. In this paper, we first recall some known examples of sets
without tangents and describe them in terms of determined directions of an
affine pointset. We show that the smallest sets without tangents in $\PG(2,5)$
are (up to projective equivalence) of two different types. We generalise the
non-trivial type by giving an explicit construction of a set without tangents
in $\PG(2,q)$, $q=p^h$, $p>2$ prime, of size $q(q-1)/2-r(q+1)/2$, for all
$0\leq r\leq (q-5)/2$. After that, a different description of the same set in
$\PG(2,5)$, using exterior sets of a conic, is given and we investigate in
which ways a set of exterior points on an external line $L$ of a conic in
$\PG(2,q)$ can be extended with an extra point $Q$ to a larger exterior set of
$\mathcal{C}$. It turns out that if $q=3$ mod 4, $Q$ has to lie on $L$, whereas
if $q=1$ mod 4, there is a unique point $Q$ not on $L$.
