Outlier detection is a statistical procedure that aims to find suspicious events or items that are different from the normal form of a dataset. It has drawn considerable interest in the field of data mining and machine learning. Outlier detection is important in many applications, including fraud detection in credit card transactions and network intrusion detection. There are two general types of outlier detection: global and local. Global outliers fall outside the normal range for an entire dataset, whereas local outliers may fall within the normal range for the entire dataset, but outside the normal range for the surrounding data points. This paper addresses local outlier detection. The best-known technique for local outlier detection is the Local Outlier Factor (LOF), a density-based technique. There are many LOF algorithms for a static data environment; however, these algorithms cannot be applied directly to data streams, which are an important type of big data. In general, local outlier detection algorithms for data streams are still deficient and better algorithms need to be developed that can effectively analyze the high velocity of data streams to detect local outliers. This paper presents a literature review of local outlier detection algorithms in static and stream environments, with an emphasis on LOF algorithms. It collects and categorizes existing local outlier detection algorithms and analyzes their characteristics. Furthermore, the paper discusses the advantages and limitations of those algorithms and proposes several promising directions for developing improved local outlier detection methods for data streams.