Random backpropagation (RBP) is a variant of the backpropagation algorithm for training neural networks, where the transpose of the forward matrices are replaced by fixed random matrices in the calculation of the weight updates. It is remarkable both because of its effectiveness, in spite of using random matrices to communicate error information, and because it completely removes the requirement of maintaining symmetric weights in a physical neural system. To better understand RBP, we compare different algorithms in terms of the information available locally to each neuron. In the process, we derive several alternatives to RBP, including skipped RBP (SRBP), adaptive RBP (ARBP), sparse RBP, and study their behavior through simulations. These simulations show that many variants are also robust deep learning algorithms, but that the derivative of the transfer function is important in the learning rule. Finally, we prove several mathematical results including the convergence to fixed points of linear chains of arbitrary length, the convergence to fixed points of linear autoencoders with decorrelated data, the long-term existence of solutions for linear systems with a single hidden layer and convergence in special cases, and the convergence to fixed points of non-linear chains, when the derivative of the activation functions is included.