Grammar error handling (GEH) is an important topic in natural language processing (NLP). GEH includes both grammar error detection and grammar error correction. Recent advances in computation systems have promoted the use of deep learning (DL) models for NLP problems such as GEH. In this survey we focus on two main DL approaches for GEH: neural machine translation models and editor models. We describe the three main stages of the pipeline for these models: data preparation, training, and inference. Additionally, we discuss different techniques to improve the performance of these models at each stage of the pipeline. We compare the performance of different models and conclude with proposed future directions.