Ultra-wideband (UWB) positioning systems often operate in a non-line-of-sight (NLOS) environment. NLOS propagation has become the main source of ultra-wideband indoor positioning errors. As such, how to identify and correct NLOS errors has become a key problem that must be solved in high-accuracy indoor positioning technology. This paper firstly describes the influence of the NLOS propagation path on localization accuracy and the generation method of ultra-wideband signals, and secondly classifies and analyzes the currently available algorithms for ultra-wideband non-line-of-sight (NLOS) identification and error suppression. For the identification of NLOS, the residual analysis judgement method, statistical feature class identification method, machine learning method and geometric feature judgement method are discussed. For the suppression of NLOS propagation errors, weighting-based methods, filtering-based methods, line-of-sight reconstruction algorithms, neural network algorithms, optimization methods with constraints, and path tracing methods are discussed. Finally, we conclude the paper and point out the problems that need to be solved in NLOS indoor positioning.