This volume contains extended abstracts outlining selected talks and other selected presentations given by participants of the workshop "Positivity and Valuations", held at the Centre de Recerca Matemàtica (CRM) in Barcelona from February 22nd to 26th, 2016. They include brief research articles reporting new results, descriptions of preliminary work or open problems, and the outcome of work in groups initiated during the workshop.

Newton-Okounkov Bodies of Exceptional Curve Plane Valuations non-Positive at Infinity.- Sufficient Conditions for the Finite Generation of Valuation Semigroups.- From Convex Geometry of Certain Valuations to Positivity Aspects in Algebraic Geometry.- Non-Positive at Infinity Valuations.- Very General Monomial Valuations on P2 and a Nagata Type Conjecture.- Valuations on Equicharacteristic Complete Noetherian Local Domains.- Desingularization by char(X)-Alterations.- Semigroup and Poincaré Series for Divisorial Valuations.- Computing Multiplier Ideals in Smooth Surfaces.- Notes On Divisors Computing MLD's and LCT's.- On the Containment Hierarchy for Simplicial Ideals.- The Universal Zeta Function for Curve Singularities and its Relation with Global Zeta Functions.- Algebraic Volumes of Divisors.- On Hirzebruch Type Inequalities and Applications.- On the Completion of Normal Toric Schemes over Rank One Valuation Rings.- Duality on Value Semigroups.- Notes on Local Positivity and Newton-Okounkov Bodies.- Newton-Okounkov Bodies and Reified Valuations of Higher Rank.