The principal aim of this thesis is to propose methods for the reconstruction of functions from noisy, random or deterministic nonequispaced data. Two of them rely on first generation wavelets. They consist in a preconditioning / interpolation on a equispaced design of the randomly designed data, folowed by wavelet shrinkage. We show that the resulting estimates are near-minimax on Holder class functions, respectively on Besov balls. We also investigate a method relying on second generation, design adapted wavelets. As a first step, yet independent, we prove that under some conditions the irregular Lagrange subdivision schemes converge and produce functions having the same number of continuous derivatives as the limit functions of the regular schemes of the same degree. Then we show the existence of design adapted multiresolution analysis and wavelet biorthogonal systems, constructed by average-interpolating subdivision. We conclude with some numerical simulations, illustrating the finite sample behaviour of the three methods.