We present CellLine, a simulator of the dynamics of gene regulatory networks (GRN) in the cells of a lineage. From user-defined reactions and initial substance quantities, it generates cell lineages, i.e. genealogic pedigrees of cells related through mitotic division. Each cells dynamics is driven by a delayed stochastic simulation algorithm (delayed SSA), allowing multiple time delayed reactions. The cells of the lineage can be individually subject to perturbations, such as gene deletion, duplication and mutation. External interventions, such as adding or removing a substance at a given moment, can be specified. Cell differentiation lineages, where differentiation is stochastically driven or externally induced, can be modeled as well. Finally, CellLine can generate and simulate the dynamics of multiple copies of any given cell of the lineage. As examples of CellLine use, we simulate the following systems: cell lineages containing a model of the P53-Mdm2 feedback loop, a differentiation lineage where each cell contains a 4 gene repressilator (a bistable circuit), a model of the differentiation of the cells of the retinal mosaic required for color vision in Drosophila melanogaster, where the differentiation pathway depends on one substances concentration that is controlled by a stochastic process, and a 9 gene GRN to illustrate the advantage of using CellLine rather than simulating multiple independent cells, in cases where the cells of the lineage are dynamically correlated.