||Reconstructing an unknown curve or surface from sample points is an important task in geometric modeling applications. Sample points obtained from real applications are usually noisy. For example, when data sets are obtained by scanning images in the plane or images in three dimensions. In computer graphics, many curve and surface reconstruction algorithms have been developed. However, their common drawback is the lack of theoretical guarantees on the quality of the reconstruction. This motivates computational geometers to propose algorithms that return provably faithful reconstructions. Algorithms of this type are known when there is no noise in the input. This leaves the problem of noise handling open. We propose a probabilistic noise model for the curve reconstruction problem. Based on this model, we design a curve reconstruction algorithm for noisy input points. The reconstruction is faithful with probability approaching 1 as the sampling density increases. Then we extend our approach to surface reconstruction from noisy input points. Not only do we improve the algorithm to make it run faster, we also make the noise model deterministic which extends its applicability and simplifies the analysis of the algorithm. We show that the surface reconstructed is faithful if the input points satisfy the deterministic noise model.