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We study the use of neural network algorithms in surface reconstruction from an unorganized point cloud, and meshing of an implicit surface. We found that for such applications, the most suitable type of neural networks is a modified version of the growing cell structure we propose here. The algorithm works by sampling randomly a target space, usually a point cloud or an implicit surface, and adjusting accordingly the neural network. The adjustment includes the connectivity of the network. Doing several experiments we found that the algorithm gives satisfactory results in some challenging situations involving sharp features and concavities. Another attractive feature of the algorithm is that its speed is virtually independent of the size of the input data, making it particularly suitable for the reconstruction of a surface from a very large point set.