Among the major developments in Mathematical Morphology in the last two decades are the interrelated subjects of connectivity classes and connected operators. Braga-Neto and Goutsias have proposed an extension of the theory of connectivity classes to a multiscale setting, whereby one can assign connectivity to an object observed at different scales. In this paper, we study connected operators in the context of multiscale connectivity. We propose the notion of a -connected operator, that is, an operator connected at scale . We devote some attention to the study of binary -grain operators. In particular, we show that families of -grain openings and -grain closings, indexed by the connectivity scale parameter, are granulometries and anti-granulometries, respectively. We demonstrate the use of multiscale connected operators with image analysis applications. The first is the scale-space representation of grayscale images using multiscale levelings, where the role of scale is played by the connectivity scale. Then we discuss the application of multiscale connected openings in granulometric analysis, where both size and connectivity information are summarized. Finally, we describe an application of multiscale connected operators to an automatic target recognition problem. 2005 Springer Science + Business Media, Inc.