  In this survey we review some important results by Gross, Hacking, Keel, and
Kontsevich on cluster algebra theory, namely, the column sign-coherence of
$C$-matrices and the Laurent positivity, both of which were conjectured by
Fomin and Zelevinsky. We digest and reconstruct the proofs of these conjectures
by Gross et al. still based on their scattering diagram method, however,
without relying on toric geometry. At the same time, we also give a detailed
account of the correspondence between the notions of cluster patterns and
scattering diagrams. Most of the results in this article are found in or
translated from the known results in the literature. However, the approach, the
construction of logic and proofs, and the overall presentation are new. Also,
as an application of the results and the techniques in the article, we show
that there is a one-to-one correspondence between $g$-vectors and cluster
variables in cluster patterns with arbitrary coefficients.
