We consider adiabatic superconducting cells operating as an artificial neuron and synapse of a multilayer perceptron (MLP). Their compact circuits contain just one and two Josephson junctions, respectively. While the signal is represented as magnetic flux, the proposed cells are inherently nonlinear and close-to-linear magnetic flux transformers. The neuron is capable of providing the one-shot calculation of sigmoid and hyperbolic tangent activation functions most commonly used in MLP. The synapse features both positive and negative signal transfer coefficients in the range ∼ (- 0.5, 0.5). We briefly discuss implementation issues and further steps toward the multilayer adiabatic superconducting artificial neural network, which promises to be a compact and the most energy-efficient implementation of MLP.