Wavelet transforms have proven very useful for a variety of signal and image processing tasks. The wedgelet transform also shows promise for certain edge-dominated images. However, in many applications we desire to introduce adaptivity and non-linearities into the transforms. These are powerful extensions, but difficult to control within the wavelet framework. The lifting scheme provides a new, spatial intuition into the wavelet transform that simplifies the introduction of adaptivity. In this thesis, we develop several new adaptive wavelet transforms and adaptive multiresolution wedgelet transforms. The lifting construction permits control over the multiresolution properties of these transforms despite the adaptivity. We demonstrate the power of our new adaptive lifted transforms with successful applications to signal denoising and compression problems.