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We propose a generalization of the synthetic controls and synthetic interventions methodology to incorporate network interference. We consider the estimation of unit-specific potential outcomes from panel data in the presence of spillover across units and unobserved confounding. Key to our approach is a novel latent factor model that takes into account network interference and generalizes the factor models typically used in panel data settings. We propose an estimator, Network Synthetic Interventions (NSI), and show that it consistently estimates the mean outcomes for a unit under an arbitrary set of counterfactual treatments for the network. We further establish that the estimator is asymptotically normal. We furnish two validity tests for whether the NSI estimator reliably generalizes to produce accurate counterfactual estimates. We provide a novel graph-based experiment design that guarantees the NSI estimator produces accurate counterfactual estimates, and also analyze the sample complexity of the proposed design. We conclude with simulations that corroborate our theoretical findings.