  The multiplicity of an algebraic curve $C$ in the complex plane at a point
$p$ on that curve is defined as the number of points that occur at the
intersection of $C$ with a general complex line that passes close to the point
$p$. It is shown that $p$ is a singular point of the curve $C$ if and only if
this multiplicity is greater than or equal to 2, in this sense, such an integer
number can be considered as a measure of how singular can be a point of the
curve $C$. In these notes, we address the classical concept of multiplicity of
singular points of complex algebraic sets (not necessarily complex curves) and
we approach the nature of the multiplicity of singular points as a geometric
invariant from the perspective of the Multiplicity Conjecture (Zariski 1971).
More precisely, we bring a discussion on the recent results obtained jointly
with Lev Birbrair, Javier Fern\'andez de Bobadilla, L\^e Dung Trang and Mikhail
Verbitsky on the bi-Lipschitz invariance of the multiplicity.
