  We consider arrangements of $n$ pseudo-lines in the Euclidean plane where
each pseudo-line $\ell_i$ is represented by a bi-infinite connected
$x$-monotone curve $f_i(x)$, $x \in \mathbb{R}$, s.t.\ for any two pseudo-lines
$\ell_i$ and $\ell_j$ with $i < j$, the function $x \mapsto f_j(x) - f_i(x)$ is
monotonically decreasing and surjective (i.e., the pseudo-lines approach each
other until they cross, and then move away from each other). We show that such
\emph{arrangements of approaching pseudo-lines}, under some aspects, behave
similar to arrangements of lines, while for other aspects, they share the
freedom of general pseudo-line arrangements.
  For the former, we prove: 1. There are arrangements of pseudo-lines that are
not realizable with approaching pseudo-lines. 2. Every arrangement of
approaching pseudo-lines has a dual generalized configuration of points with an
underlying arrangement of approaching pseudo-lines.
  For the latter, we show: 1. There are $2^{\Theta(n^2)}$ isomorphism classes
of arrangements of approaching pseudo-lines (while there are only $2^{\Theta(n
\log n)}$ isomorphism classes of line arrangements). 2. It can be decided in
polynomial time whether an allowable sequence is realizable by an arrangement
of approaching pseudo-lines.
  Furthermore, arrangements of approaching pseudo-lines can be transformed into
each other by flipping triangular cells, i.e., they have a connected flip
graph, and every bichromatic arrangement of this type contains a bichromatic
triangular cell.
