We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties ( X , L ) (X,L) ; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope μ \mu for varieties and their subschemes; if ( X , L ) (X,L) is semistable, then μ ( Z ) ≤ μ ( X ) \mu (Z)\le \mu (X) for all Z ⊂ X Z\subset X . We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.