The inverse sum indeg index is a recently-introduced bond-additive descriptor that was selected by Vukicevic and Gasperov (2010) as a significant predictor of total surface area of octane isomers. In this paper, we present several sharp upper and lower bounds on the inverse sum indeg index in terms of some graph parameters such as the order, size, radius, number of pendant vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree, and relate this index to various well-known graph invariants such as the ordinary and multiplicative Zagreb indices, Randic indices, sum-connectivity index, modified Zagreb index, harmonic index, forgotten index, and eccentric connectivity index.