Low-frequency 1/f noise is ubiquitous and dominates the signal-to-noise performance in nanodevices. Here we investigate the noise characteristics of single-layer and bilayer graphene nanodevices and uncover an unexpected 1/f noise behavior for bilayer devices. Graphene is a single layer of graphite, where carbon atoms form a two-dimensional (2D) honeycomb lattice. Despite the similar composition, bilayer graphene (two graphene monolayers stacked in the natural graphite order) is a distinct 2D system with a different band structure and electrical properties. 1,2In graphene monolayers, the 1/f noise is found to follow Hooge's empirical relation with a noise parameter comparable to that of bulk semiconductors. However, this 1/f noise is strongly suppressed in bilayer graphene devices and exhibits an unusual dependence on the carrier density, different from most other materials. The unexpected noise behavior in graphene bilayers is associated with its unique band structure that varies with the charge distribution among the two layers, resulting in an effective screening of potential fluctuations due to external impurity charges. The findings here point to exciting opportunities for graphene bilayers in low-noise applications.