In living cells, DNA is packaged along with protein and RNA into chromatin. Chemical modifications to nucleotides and histone proteins are added, removed and recognized by multi-functional molecular complexes. Here I define a new computational model, in which chromatin modifications are information units that can be written onto a one-dimensional string of nucleosomes, analogous to the symbols written onto cells of a Turing machine tape, and chromatin-modifying complexes are modeled as read-write rules that operate on a finite set of adjacent nucleosomes. I illustrate the use of this "chromatin computer" to solve an instance of the Hamiltonian path problem. I prove that chromatin computers are computationally universal--and therefore more powerful than the logic circuits often used to model transcription factor control of gene expression. Features of biological chromatin provide a rich instruction set for efficient computation of nontrivial algorithms in biological time scales. Modeling chromatin as a computer shifts how we think about chromatin function, suggests new approaches to medical intervention, and lays the groundwork for the engineering of a new class of biological computing machines.