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Identification of treatment effects in the presence of unmeasured confounding is a persistent problem in the social, biological, and medical sciences. The problem of unmeasured confounding in settings with multiple treatments is most common in statistical genetics and bioinformatics settings, where researchers have developed many successful statistical strategies without engaging deeply with the causal aspects of the problem. Recently there have been a number of attempts to bridge the gap between these statistical approaches and causal inference, but these attempts have either been shown to be flawed or have relied on fully parametric assumptions. In this paper, we propose two strategies for identifying and estimating causal effects of multiple treatments in the presence of unmeasured confounding. The auxiliary variables approach leverages variables that are not causally associated with the outcome; in the case of a univariate confounder, our method only requires one auxiliary variable, unlike existing instrumental variable methods that would require as many instruments as there are treatments. An alternative null treatments approach relies on the assumption that at least half of the confounded treatments have no causal effect on the outcome, but does not require a priori knowledge of which treatments are null. Our identification strategies do not impose parametric assumptions on the outcome model and do not rest on estimation of the confounder. This paper extends and generalizes existing work on unmeasured confounding with a single treatment and models commonly used in bioinformatics.