Spectrally sparse data arise in many areas of science and engineering, for instance magnetic resonance imaging, fluorescence microscopy and radar imaging. Spectral compressed sensing is about reconstructing spectrally sparse data from incomplete information. Two difference classes of nonconvex optimization algorithms are introduced to tackle this problem. They are developed by exploiting low rank structure within the data in two different ways: one is based on the embedded manifold of low rank matrices and the other is based on the factorization model of low rank matrices. Theoretical recovery guarantees will be presented for the proposed algorithms under certain random models, showing that the sampling complexity is essentially proportional to the intrinsic dimension of the problems rather the ambient dimension. Empirical observations demonstrate the efficacy of the algorithms. Joint with Jian-Feng Cai (HKUST) and Tianming Wang (UT Austin).