We present a novel form of interactive video object segmentation where a few clicks by the user helps the system produce a full spatio-temporal segmentation of the object of interest. Whereas conventional interactive pipelines take the user's initialization as a starting point, we show the value in the system taking the lead even in initialization. In particular, for a given video frame, the system precomputes a ranked list of thousands of possible segmentation hypotheses (also referred to as object region proposals) using image and motion cues. Then, the user looks at the top ranked proposals, and clicks on the object boundary to carve away erroneous ones. This process iterates (typically 2-3 times), and each time the system revises the top ranked proposal set, until the user is satisfied with a resulting segmentation mask. Finally, the mask is propagated across the video to produce a spatio-temporal object tube. On three challenging datasets, we provide extensive comparisons with both existing work and simpler alternative methods. In all, the proposed Click Carving approach strikes an excellent balance of accuracy and human effort. It outperforms all similarly fast methods, and is competitive or better than those requiring 2 to 12 times the effort.