The Cassie-Baxter model is widely used to predict the apparent contact angles obtained on composite (solid-liquid-air) superhydrophobic interfaces. However, the validity of this model has been repeatedly challenged by various research groups because of its inherent inability to predict contact angle hysteresis. In our recent work, we have developed robust omniphobic surfaces which repel a wide range of liquids. An interesting corollary of constructing such surfaces is that it becomes possible to directly image the solid-liquid-air triple-phase contact line on a composite interface, using an electron microscope with non-volatile organic liquids or curable polymers. Here, we fabricate a range of model superoleophobic surfaces with controlled surface topography in order to correlate the details of the local texture with the experimentally observed apparent contact angles. Based on these experiments, in conjunction with numerical simulations, we modify the classical Cassie-Baxter relation to include a local differential texture parameter which enables us to quantitatively predict the apparent advancing and receding contact angles, as well as contact angle hysteresis. This quantitative prediction also allows us to provide an a priori estimation of roll-off angles for a given textured substrate. Using this understanding we design model substrates that display extremely small or extremely large roll-off angles, as well as surfaces that demonstrate direction-dependent wettability, through a systematic control of surface topography and connectivity. (C) 2009 Elsevier Inc. All rights reserved.