Utilizing graph algorithms is a common activity in computer science. Algorithms that perform computations on large graphs are not always efficient. This work investigates the Single-Source Shortest Path (SSSP) problem, which is considered to be one of the most important and most studied graph problems. This thesis contains a review of the SSSP problem in both theory and practice. In addition, it discusses a new single-source shortest-path algorithm that achieves the same $O(n \cdot m)$ time bound as the traditional Bellman-Ford-Moore algorithm but outperforms it and other state-of-the-art algorithms in practice. The work is comprised of three parts. The first discusses some basic shortest-path and negative-cycle-detection algorithms in literature from the theoretical and practical point of view. The second contains a discussion of a new algorithm for the single-source shortest-path problem that outperforms most state-of-the-art algorithms for several well-known families of graphs. The main idea behind the proposed algorithm is to select the fewest most-effective vertices to scan. We also propose a discussion of correctness, termination, and the proof of the worst-case time bound of the proposed algorithm. This section also suggests two different implementations for the proposed algorithm, the first runs faster while the second performs a fewer number of operations. Finally, an extensive computational study of the different shortest paths algorithms is conducted. The results are proposed using a new evaluation metric for shortest-path algorithms. A discussion of outcomes, strengths, and weaknesses of the various shortest path algorithms are also included in this work.