Let X X be a complete variety of dimension n n over an algebraically closed field K \mathbb {K} . Let V ∙ V_\bullet be a graded linear series associated to a line bundle L L on X X , that is, a collection { V m } m ∈ N \{V_m\}_{m\in \mathbb {N}} of vector subspaces V m ⊆ H 0 ( X , L ⊗ m ) V_m\subseteq H^0(X,L^{\otimes m}) such that V 0 = K V_0=\mathbb {K} and V k ⋅ V ℓ ⊆ V k + ℓ V_k\cdot V_\ell \subseteq V_{k+\ell } for all k , ℓ ∈ N k,\ell \in \mathbb {N} . For each m m in the semigroup \[ N ( V ∙ ) = { m ∈ N ∣ V m ≠ 0 } , \mathbf {N}(V_\bullet )=\{m\in \mathbb {N}\mid V_m\ne 0\}, \] the linear series V m V_m defines a rational map \[ ϕ m : X ⇢ Y m ⊆ P ( V m ) , \phi _m\colon X\dashrightarrow Y_m\subseteq \mathbf {P}(V_m), \] where Y m Y_m denotes the closure of the image ϕ m ( X ) \phi _m(X) . We show that for all sufficiently large m ∈ N ( V ∙ ) m\in \mathbf {N}(V_\bullet ) , these rational maps ϕ m : X ⇢ Y m \phi _m\colon X\dashrightarrow Y_m are birationally equivalent, so in particular Y m Y_m are of the same dimension κ \kappa , and if κ = n \kappa =n then ϕ m : X ⇢ Y m \phi _m\colon X\dashrightarrow Y_m are generically finite of the same degree. If N ( V ∙ ) ≠ { 0 } \mathbf {N}(V_\bullet )\ne \{0\} , we show that the limit \[ v o l κ ( V ∙ ) = lim m ∈ N ( V ∙ ) dim K V m m κ / κ ! vol_\kappa (V_\bullet )=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\dim _\mathbb {K} V_m}{m^\kappa /\kappa !} \] exists, and 0 > v o l κ ( V ∙ ) > ∞ 0>vol_\kappa (V_\bullet )>\infty . Moreover, if Z ⊆ X Z\subseteq X is a general closed subvariety of dimension κ \kappa , then the limit \[ ( V ∙ κ ⋅ Z ) m o v = lim m ∈ N ( V ∙ ) # ( ( D m , 1 ∩ ⋯ ∩ D m , κ ∩ Z ) ∖ B s ( V m ) ) m κ (V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\#\bigl ((D_{m,1}\cap \cdots \cap D_{m,\kappa }\cap Z)\setminus Bs(V_m)\bigr )}{m^\kappa } \] exists, where D m , 1 , … , D m , κ ∈ | V m | D_{m,1},\ldots ,D_{m,\kappa }\in |V_m| are general divisors, and \[ ( V ∙ κ ⋅ Z ) m o v = deg ( ϕ m | Z : Z ⇢ ϕ m ( Z ) ) v o l κ ( V ∙ ) (V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\deg \bigl (\phi _m|_Z\colon Z\dashrightarrow \phi _m(Z)\bigr )vol_\kappa (V_\bullet ) \] for all sufficiently large m ∈ N ( V ∙ ) m\in \mathbf {N}(V_\bullet ) .