An essential feature of all practical de novo molecule generating programs is the ability to focus the potential combinatorial explosion of grown molecules on a desired chemical space. It is a daunting task to balance the generation of new molecules with limitations on growth that produce desired features such as stability in water, synthetic accessibility, or drug-likeness. We have developed an algorithm, Fragment Optimized Growth (FOG), which statistically biases the growth of molecules with desired features. At the heart of the algorithm is a Markov Chain which adds fragments to the nascent molecule in a biased manner, depending on the frequency of specific fragment-fragment connections in the database of chemicals it was trained on. We show that in addition to generating synthetically feasible molecules, it can be trained to grow new molecules that resemble desired classes of molecules such as drugs, natural products, and diversity-oriented synthetic products. In order to classify our grown molecules, we developed the Topology Classifier (TopClass) algorithm that is capable of classifying compounds, for example as drugs or nondrugs. The classification accuracies obtained with TopClass compare favorably with the literature. Furthermore, in contrast to "black-box" approaches such as Neural Networks, TopClass brings to light characteristics of drugs that distinguish them from nondrugs.