1/f noise and statistically self-similar processes such as fractional Brownian motion (fBm) are vital for modeling numerous real-world phenomena, from network traffic to DNA to the stock market. Although several algorithms exist for synthesizing discrete-time samples of a 1/f process, these algorithms are inexact, meaning that the covariance of the synthesized processes can deviate significantly from that of a true 1/f process. However, the Fast Fourier Transform (FFT) can be used to exactly and efficiently synthesize such processes in O(N logN) operations for a length-N signal. Strangely enough, the key is to apply the FFT to match the target process's covariance structure, not its frequency spectrum. In this paper, we prove that this FFT-based synthesis is exact not only for 1/f processes such as fBm, but also for a wide class of long-range dependent processes. Leveraging the flexibility of the FFT approach, we develop new models for processes that exhibit one type of fBm scaling behavior over fine resolutions and a distinct scaling behavior over coarse resolutions. We also generalize the method in order to exactly synthesize various nonGaussian 1/f processes. Our nonGaussian 1/f synthesis is fast and simple. Used in simulations, our synthesis techniques could lead to new insights into areas such as computer networking, where the traffic processes exhibit nonGaussianity and a richer covariance than that of a strict fBm process.