Standard wavelet linear approximations generate oscillations (Gibbs' phenomenon) near singularities in piecewise smooth functions. Nonlinear and data dependent methods are often considered as the main strategies to avoid those oscillations. Using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing to standard wavelet transforms, we have designed an adaptive ENO-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining arbitrary high order accuracy uniformly and having a multiresolution framework and fast algorithms, all without any edge artifacts . We have also shown the stability of the ENO-wavelet transforms and obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We briefly discuss several applications of the ENO-wavelet transforms, including function approximation, image compression and signal denoising.