Nonconvex optimization problems are the bottleneck in many applications in science and technology. Often these problems are NP-hard and they are approached with ad hoc methods that frequently fail to yield satisfactory results. This issue becomes even more prevalent in the ``Big Data Regime''. In my talk I will report on recent breakthroughs in solving some important nonconvex optimization problems. In particular, I will discuss the problems of phase retrieval, blind deconvolution, and blind source separation. The most notorious among these three is arguably phase retrieval, which is the century-old problem of reconstructing a function from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems arise in numerous areas including X-ray crystallography, differential geometry, astronomy, diffraction imaging, and quantum physics and are very difficult to solve numerically. Combining tools from optimization, random matrix theory and harmonic analysis, we will derive rigorous mathematical methods that can solve the aforementioned problems under meaningful practical conditions. The proposed methods come with rigorous theoretical guarantees, are numerically efficient and stable in the presence of noise, and require little or no parameter tuning, thus making them useful for massive datasets. I will also discuss connections to the emerging field of self-calibration, which is based on the idea of equipping a sensor with a smart algorithm that can compensate automatically for the sensor's imperfections. The effectiveness of our methods will be illustrated in applications such as astronomy, X-ray crystallography, terahertz imaging, and the Internet-of-Things.