Zero-order and twin images are a serious obstacle in achieving a high-quality output in in-line digital holography (DH). They decrease the useful bandwidth of the off-axis DH. Over the years the twin image removal problem was approached both by instrumental and numerical means. The paper provides an extended survey of the proposed solutions with their pros and cons as a guide for further advance in this field. Processing of a single spatial carrier fringe pattern involves spatial filtering in the frequency domain, spatial phase-shifting (PS) or wavelet transform. A point source digital holographic microscopy (DHM), introduction of calibration measurements or various modifications of PS technique are instrumental solutions to the twin image problem for in-line DH. Numerical solutions to the same problem include iterative and non-iterative approaches, diffraction-based and inverse problem solutions, reconstruction of purely real or phase objects and of complex objects, reconstruction of plane and volume objects. Elimination only of the zero-order image relies on non-linear filtering or additional calibration measurements.