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.
Comment: v3: 32 pages; Proposition 3.6 and Corollary 4.5 added; this has simplified several results in Section 3 and allowed us to show every stratifying system is part of a minimal projective one in Section 4; a missing assumption added to Theorem 3.19(=Theorem C); other minor changes. v2: 32 pages; Remark 5.3 added; typos corrected; other minor changes. v1: 31 pages. Comments very welcome!