We propose a definition of “nonabelian mixed Hodge structure” together with a construction associating to a smooth projective variety X and to a nonabelian mixed Hodge structure V , the “nonabelian cohomology of X with coefficients in V ” which is a (pre-)nonabelian mixed Hodge structure denoted H = Hom(XM , V ). We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where V has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with P. For this example we show that Hom(XM , V ) is a namhs for any smooth projective variety X. Introduction—p. 2 Part I: Nonabelian weight filtrations Conventions—p. 14 Nonabelian filtrations—p. 15 Perfect complexes and Dold-Puppe linearization—p. 22 Further study of filtered n-stacks—p. 26 Filtered and Gm-equivariant perfect complexes—p. 33 Weight-filtered n-stacks—p. 38 Analytic and real structures—p. 43 Part II: The main definitions and conjectures Pre-nonabelian mixed Hodge structures—p. 50 Mixed Hodge complexes and linearization—p. 52 Nonabelian mixed Hodge structures—p. 59 Homotopy group sheaves—p. 61 The basic construction—p. 75 The basic conjectures—p. 95 Variations of nonabelian mixed Hodge structure—p. 98 Part III: Computations Some morphisms between Eilenberg-MacLane pre-namhs—p. 106 Construction of a namhs V of homotopy type S C —p. 111 The namhs on cohomology of a smooth projective variety with coefficients in V —p. 113 University of California at Irvine, Irvine, CA 92697, USA. Partially supported by NSF Career Award DMS-9875383 and A.P. Sloan research fellowship. University of Pennsylvania, 209 South 33rd Street Philadelphia, PA 19104-6395, USA. Partially supported by NSF Grant DMS-9800790 and A.P. Sloan Research Fellowship. CNRS, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France.