This paper is devoted to the study of multigraded algebras and multigraded linear series. For an N s -graded algebra A, we define and study its volume function F A : N s + → R, which computes the asymptotics of the Hilbert function of A. We relate the volume function F A to the volume of the fibers of the global Newton-Okounkov body ∆(A) of A. Unlike the classical case of standard multigraded algebras, the volume function F A is not a polynomial in general. However, in the case when the algebra A has a decomposable grading, we show that the volume function F A is a polynomial with non-negative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series.Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties. CONTENTS 1. Introduction 2. Notation and preliminaries 3. Singly graded algebras of almost integral type 3.1. The case where R is a domain 3.2. Valuations with bounded leaves 4. Multigraded algebras of almost integral type 4.1. Global Newton-Okounkov bodies 5. Multigraded algebras of almost integral type with decomposable grading 6. Application to graded families of ideals 6.1. Positivity for graded families of equally generated ideals 6.2. Positivity for mixed volumes of convex bodies 7. Multigraded linear series References