With the variety of measurement techniques available on the market today, fusing multi-source complementary information into one dataset is a matter of great interest. Target-based, point-based and feature-based methods are some of the approaches used to place data in a common reference frame by estimating its corresponding transformation parameters. This paper proposes a new linear feature-based method to perform accurate registration of point clouds, either in 2D or 3D. A two-step fast algorithm called Robust Line Matching and Registration (RLMR), which combines coarse and fine registration, was developed. The initial estimate is found from a triplet of conjugate line pairs, selected by a RANSAC algorithm. Then, this transformation is refined using an iterative optimization algorithm. Conjugates of linear features are identified with respect to a similarity metric representing a line-to-line distance. The efficiency and robustness to noise of the proposed method are evaluated and discussed. The algorithm is valid and ensures valuable results when pre-aligned point clouds with the same scale are used. The studies show that the matching accuracy is at least 99.5%. The transformation parameters are also estimated correctly. The error in rotation is better than 2.8% full scale, while the translation error is less than 12.7%.