In this thesis is in detail explained work with one-layer neural network with backpropagation. During the neural network learning process user can choose between four activation functions and two momentums. Also, user have the opportunity to write the wanted number of hidden layer neurons, the amount of learning speed, the amount of NRMS and number of learning steps. The number of input and output neurons are defined by single task learning. The learning results user can get in a tabular format or with graphic display. The user can test the neural networks with given numbers and with numbers that are not defined in the learning process. User can check neural networks operations if he use structure with two inputs and one, three or four outputs. Inputs in the learning process are numbers between one and five, and their combinations user selects on the start, and after that user can select neural network learning parameters. Outputs from neural network are results of different mathematical operations for the default input combinations of numbers. Work principle and neural network learning process is explained in the first part of this thesis, while examples of work and testing of the neural networks are shown in the second part. At the end of paper is shown and explained the work of two-layer neural network with backpropagation, also with two inputs and with one and four outputs. The graphical interface for each example is made in MATLAB.