Quark and gluon condensates in nuclear matter are studied. These in-medium condensates may be linked to a wide range of nuclear phenomena and are important inputs to QCD sum-rule calculations at finite density. The Hellmann-Feynman theorem yields a prediction of the quark condensate that is model independent to first order in the nucleon density. This linear density dependence, with slope determined by the nucleon σ term, implies that the quark condensate is reduced considerably at nuclear matter saturation density-it is roughly 25-50 % smaller than the vacuum value. The trace anomaly and the Hellmann-Feynman theorem lead to a prediction of the gluon condensate that is model independent to first order in the nucleon density. At nuclear matter saturation density, the gluon condensate is about 5% smaller than the vacuum value. Contributions to the in-medium quark condensate that are of higher order in the nucleon density are estimated with mean-field quark-matter calculations using the Nambu-Jona-Lasinio and Gell-Mann-Lévy models. Treatments of nuclear matter based on hadronic degrees of freedom are also considered, and the uncertainties are discussed.