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.