In observational studies, continuous exposures, such as lead exposure and air pollution, are prevalent. At the same time, unmeasured confounding is a major concern in observational studies, as the treatment assignment is not controlled by the scientist as in a randomized experiment. A common approach to assuage concerns about unmeasured confounding is through a sensitivity analysis, which asks how strong the unmeasured confounding must be to overturn a qualitative causal conclusion. This thesis introduces three methods aimed at conducting sensitivity analysis with a continuous exposure. The sensitivity models considered allow for the unmeasured confounder to have a bounded impact on the exposure. The first chapter presents a new sensitivity analysis framework for matched observational studies when the exposure is continuous and the outcome is binary, focusing on a special class of test statistics. The second chapter extends the sensitivity analysis framework from the first to continuous outcomes and accommodates arbitrary test statistics. Both chapters develop methods for testing sharp and weak null hypotheses, and are agnostic to the type of matched design. The third chapter derives bounds on the average derivative effect under a related sensitivity model that constrains the odds ratio (at any two dose levels) between the latent and observed generalized propensity score. Flexible, efficient estimators for the bounds and corresponding confidence intervals are derived.