Artificially intelligent algorithmic systems have the potential to transform large sectors of the economy, and the insurance sector makes no exception. Although the insurance industry might have been slow in recognizing and exploiting the value of artificial intelligence at first, today the applications of intelligent machines in insurance are growing. This paper considers the role that artificial intelligence and machine learning can play in insurance. It discusses some current uses and also focuses on the possible application of artificial intelligence to smart contracts. Some of the main issues related to artificially intelligent algorithmic systems are also touched upon, highlighting the need to ensure transparency and accountability of automated decision-making.