Neural networks are making headlines every day as the tool of the future, powering artificial intelligence programs and supporting technologies never seen before. However, the training of neural networks can take days or even weeks for bigger networks, and requires the use of super computers and GPUs in academia and industry in order to achieve state of the art results. This thesis discusses employing selective measures to determine when to backpropagate and forward propagate in order to reduce training time while maintaining classification performance. This thesis tests these new algorithms on the MNIST and CASIA datasets, and achieves successful results with both algorithms on the two datasets. The selective backpropagation algorithm shows a reduction of up to 93.3% of backpropagations completed, and the selective forward propagation algorithm shows a reduction of up to 72.90% in forward propagations and backpropagations completed compared to baseline runs of always forward propagating and backpropagating. This work also discusses employing the selective backpropagation algorithm on a modified dataset with disproportional under-representation of some classes compared to others.