The training of two-layer neural networks with nonlinear activation functions is an important non-convex optimization problem with numerous applications and promising performance in layerwise deep learning. In this paper, we develop exact convex optimization formulations for two-layer neural networks with second degree polynomial activations based on semidefinite programming. Remarkably, we show that semidefinite lifting is always exact and therefore computational complexity for global optimization is polynomial in the input dimension and sample size for all input data. The developed convex formulations are proven to achieve the same global optimal solution set as their non-convex counterparts. More specifically, the globally optimal two-layer neural network with polynomial activations can be found by solving a semidefinite program (SDP) and decomposing the solution using a procedure we call Neural Decomposition. Moreover, the choice of regularizers plays a crucial role in the computational tractability of neural network training. We show that the standard weight decay regularization formulation is NP-hard, whereas other simple convex penalties render the problem tractable in polynomial time via convex programming. We extend the results beyond the fully connected architecture to different neural network architectures including networks with vector outputs and convolutional architectures with pooling. We provide extensive numerical simulations showing that the standard backpropagation approach often fails to achieve the global optimum of the training loss. The proposed approach is significantly faster to obtain better test accuracy compared to the standard backpropagation procedure.