Neutral landscapes and mutational robustness are believed to be important enablers of evolvability in biology. We apply these concepts to software, defining mutational robustness to be the fraction of random mutations that leave a program's behavior unchanged. Test cases are used to measure program behavior and mutation operators are taken from genetic programming. Although software is often viewed as brittle, with small changes leading to catastrophic changes in behavior, our results show surprising robustness in the face of random software mutations. The paper describes empirical studies of the mutational robustness of 22 programs, including 14 production software projects, the Siemens benchmarks, and 4 specially constructed programs. We find that over 30% of random mutations are neutral with respect to their test suite. The results hold across all classes of programs, for mutations at both the source code and assembly instruction levels, across various programming languages, and are only weakly related to test suite coverage. We conclude that mutational robustness is an inherent property of software, and that neutral variants (i.e., those that pass the test suite) often fulfill the program's original purpose or specification. Based on these results, we conjecture that neutral mutations can be leveraged as a mechanism for generating software diversity. We demonstrate this idea by generating a population of neutral program variants and showing that the variants automatically repair unknown bugs with high probability. Neutral landscapes also provide a partial explanation for recent results that use evolutionary computation to automatically repair software bugs.