Title: Biologically Plausible Neural Networks via Evolutionary Dynamics and Dopaminergic Plasticity. Abstract: Artificial neural networks (ANNs) lack in biological plausibility, chiefly because backpropagation requires a variant of plasticity (precise changes of the synaptic weights informed by neural events that occur downstream in the neural circuit) that is profoundly incompatible with the current understanding of the animal brain. Here we propose that backpropagation can happen in evolutionary time, instead of lifetime, in what we call neural net evolution (NNE). In NNE the weights of the links of the neural net are sparse linear functions of the animal's genes, where each gene has two alleles, 0 and 1. In each generation, a population is generated at random based on current allele frequencies, and it is tested in the learning task. The relative performance of the two alleles of each gene over the whole population is determined, and the allele frequencies are updated via the standard population genetics equations for the weak selection regime. We prove that, under assumptions, NNE succeeds in learning simple labeling functions with high probability, and with polynomially many generations and individuals per generation. We test the NNE concept, with only one hidden layer, on MNIST with encouraging results. Finally, we explore a further version of biologically plausible ANNs inspired by the recent discovery in animals of dopaminergic plasticity: the increase of the strength of a synapse that fired if dopamine was released soon after the firing.