  We investigate finite 3-nets embedded in a projective plane over a (finite or
infinite) field of any characteristic p. Such an embedding is regular when each
of the three classes of the 3-net comprises concurrent lines, and irregular
otherwise. It is completely irregular when no class of the 3-net consists of
concurrent lines. We are interested in embeddings of 3-nets which are irregular
but the lines of one class are concurrent. For an irregular embedding of a
3-net of order n greater than 4 we prove that, if all lines from two classes
are tangent to the same irreducible conic, then all lines from the third class
are concurrent. We also prove the converse provided that the order n of the
3-net is smaller than p. In the complex plane, apart from a sporadic example of
order n=5 due to Stipins, each known irregularly embedded 3-net has the
property that all its lines are tangent to a plane cubic curve. Actually, the
procedure of constructing irregular 3-nets with this property works over any
field. In positive characteristic, we present some more examples for n greater
than 4 and give a complete classification for n=4.
