Dimensionality reduction techniques play important roles in the analysis of big data. Traditional dimensionality reduction approaches, such as principal component analysis (PCA) and linear discriminant analysis (LDA), have been studied extensively in the past few decades. However, as the dimensionality of data increases, the computational cost of traditional dimensionality reduction methods grows exponentially, and the computation becomes prohibitively intractable. These drawbacks have triggered the development of random projection (RP) techniques, which map high-dimensional data onto a low-dimensional subspace with extremely reduced time cost. However, the RP transformation matrix is generated without considering the intrinsic structure of the original data and usually leads to relatively high distortion. Therefore, in recent years, methods based on RP have been proposed to address this problem. In this paper, we summarize the methods used in different situations to help practitioners to employ the proper techniques for their specific applications. Meanwhile, we enumerate the benefits and limitations of the various methods and provide further references for researchers to develop novel RP-based approaches.