Although the Discrete Wavelet Transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages. First, the DWT is shift sensitive because input-signal shifts generate unpredictable changes in DWT coefficients. Second, the DWT suffers from poor directionality because DWT coefficients reveal only three spatial orientations. Third, DWT analysis lacks the phase information that accurately describes non-stationary signal behavior. To overcome these disadvantages, we introduce the notion of projection-based complex wavelet transforms. These two-stage, projection-based complex wavelet transforms consist of a projection onto a complex function space followed by a DWT of the complex projection. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the projection stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. We exploit this flexibility to create the Complex Double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of ^{(3^m - 1)}/_{(2^m - 1)} in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy. Besides the mitigation of DWT shortcomings, our transforms have unique properties that will potentially benefit a variety of signal processing applications. As an example, we demonstrate that our projection-based complex wavelet transforms achieve state-of-the-art results in a seismic signal-processing application.