Today, the Internet is an effective channel for social interaction worldwide, but it also opens up great opportunities for cyberattacks. Recently, the number of botnets and phishing cyberattacks has increased dramatically. Phishing is a cyber attack carried out using a fake website designed to steal sensitive information. Detecting such attacks using machine learning algorithms and neural systems is an efficient and fast method. This paper discusses the types of Internet threats and neural network algorithms used to counter phishing attacks. The main goal of this paper is to build neural network models capable of effectively detecting phishing attacks. The methods of research are data analysis, data preprocesses and neural networks. In the first step, the Kaggle dataset containing data from 5000 legitimate and 5000 malicious web pages was gathered. The features of the dataset were analyzed and normalized. Then the Deep neural network, Convolutional neural network and Long short-term memory network were applied for threat classification. The Deep neural network and Long short-term memory network showed the best results of accuracy up to 0.96 and 0.99, while the Convolutional neural network reached only 0.75. Generally, the Deep neural network and Long short-term memory achieved the best results among the neural network methods.