Two-dimensional (2D) materials are at the heart of many novel devices due to their unique and often superior properties. For simplicity, 2D materials are often assumed to exist in their text-book form, i.e., as an ideal solid with no imperfections. However, defects are ubiquitous in macroscopic samples and play an important – if not imperative – role for the performance of any device. Thus, many independent studies have targeted the artificial introduction of defects into 2D materials by particle irradiation. In our view it would be beneficial to develop general defect engineering strategies for 2D materials based on a thorough understanding of the defect creation mechanisms, which may significantly vary from the ones relevant for 3D materials. This paper reviews the state-of-the-art in defect engineering of 2D materials by electron and ion irradiation with a clear focus on defect creation on the atomic scale and by individual impacts. Whenever possible we compile reported experimental data alongside corresponding theoretical studies. We show that, on the one hand, defect engineering by particle irradiation covers a wide range of defect types that can be fabricated with great precision in the most commonly investigated 2D materials. On the other hand, gaining a complete understanding still remains a challenge, that can be met by combining advanced theoretical methods and improved experimental set-ups, both of which only now begin to emerge. In conjunction with novel 2D materials, this challenge promises attractive future opportunities for researchers in this field.