  Stratifying systems, which have been defined for module, triangulated and
exact categories previously, were developed to produce examples of standardly
stratified algebras. A stratifying system $\Phi$ is a finite set of objects
satisfying some orthogonality conditions. One very interesting property is that
the subcategory $\mathcal{F}(\Phi)$ of objects admitting a composition
series-like filtration with factors in $\Phi$ has the Jordan-H{\"{o}}lder
property on these filtrations.
  This article has two main aims. First, we introduce notions of subobjects,
simple objects and composition series for an extriangulated category, in order
to define a Jordan-H{\"{o}}lder extriangulated category. Moreover, we
characterise Jordan-H{\"{o}}lder, length, weakly idempotent complete
extriangulated categories in terms of the associated Grothendieck monoid and
Grothendieck group. Second, we develop a theory of stratifying systems in
extriangulated categories. We define projective stratifying systems and show
that every stratifying system $\Phi$ in an extriangulated category is part of a
minimal projective one $(\Phi,Q)$. We prove that $\mathcal{F}(\Phi)$ is a
Jordan-H{\"{o}}lder extriangulated category when $(\Phi,Q)$ satisfies a left
exactness condition.
