  We study point-line incidence structures and their properties in the
projective plane. Our motivation is the problem of the existence of $(n_4)$
configurations, still open for few remaining values of $n$. Our approach is
based on quasi-configurations: point-line incidence structures where each point
is incident to at least $3$ lines and each line is incident to at least $3$
points. We investigate the existence problem for these quasi-configurations,
with a particular attention to $3|4$-configurations where each element is $3$-
or $4$-valent. We use these quasi-configurations to construct the first
$(37_4)$ and $(43_4)$ configurations. The existence problem of finding
$(22_4)$, $(23_4)$, and $(26_4)$ configurations remains open.
