Getting realistic theories of inflation from string theory is not easy. Recent progress has focused on flux compactifications of type IIB string theory containing anti-D3 branes to lift AdS minima to de Sitter space. I describe two recent works in this context: brane-antibrane inflation in singular compactifications, which allow for realistic chiral matter, and 'racetrack inflation', where moduli of the Calabi-Yau manifold serve as the inflaton, realizing an example of topological inflation. Emphasis will be given to observable signatures in the cosmic microwave background. I also discuss a paradigm for reheating after brane-antibrane inflation, where the observable universe arises dynamically as a 3-brane, which forms at the end of inflation.