This study reports the results of the first systematic investigation into Astro 101 students' conceptual and reasoning difficulties with cosmology. We developed four surveys with which we measured students' conceptual knowledge of the Big Bang, the expansion and evolution of the universe, and the evidence for dark matter. Our classical test theory and item response theory analyses of over 2300 students' pre- and post-instruction responses, combined with daily classroom observations, videotapes of students working in class, and one-on-one semi-structured think-aloud interviews with nineteen Astro 101 students, revealed several common learning difficulties. In order to help students overcome these difficulties, we used our results to inform the development of a new suite of cosmology lecture-tutorials. In our initial testing of the new lecture-tutorials at the University of Colorado at Boulder and the University of Arizona, we found many cases in which students who used the lecture-tutorials achieved higher learning gains (as measured by our surveys) at statistically significant levels than students who did not. Subsequent use of the lecture-tutorials at a variety of colleges and universities across the United States produced a wide range of learning gains, suggesting that instructors' pedagogical practices and implementations of the lecture-tutorials significantly affect whether or not students achieve high learning gains.