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On-demand ride-sharing services such as Uber and Lyft have gained tremendous popularity over the past decade, largely driven by the omnipresence of mobile devices. Ride-sharing services can provide economic and environmental benefits such as reducing traffic congestion and vehicle emissions. Multi-hop ride-sharing enables passengers to transfer between vehicles within a single trip, which significantly extends the benefits of ride-sharing and provides ride opportunities that are not possible otherwise. Despite its advantages, offering real-time multi-hop ride-sharing services at large scale is a challenging computational task due to the large combination of vehicles and passenger transfer points. To address these challenges, we propose exact and approximation algorithms that are scalable and achieve real-time responses for highly dynamic ride-sharing scenarios in large metropolitan areas. Our experiments on real-world datasets show the benefits of multi-hop ride-sharing services and demonstrate that our proposed algorithms are more than two orders of magnitude faster than the state-of-the-art. Our approximation algorithms offer a comparable trip quality to our exact algorithm, while improving the ride-sharing request matching time by another order of magnitude.