We address the task of auto-completing data cells in relational tables. Such tables describe entities (in rows) with their attributes (in columns). We present the CellAutoComplete framework to tackle several novel aspects of this problem, including: (i) enabling a cell to have multiple, possibly conflicting values, (ii) supplementing the predicted values with supporting evidence, (iii) combining evidence from multiple sources, and (iv) handling the case where a cell should be left empty. Our framework makes use of a large table corpus and a knowledge base as data sources, and consists of preprocessing, candidate value finding, and value ranking components. Using a purpose-built test collection, we show that our approach is 40\% more effective than the best baseline.