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We report the discovery of ten new pulsars in the globular cluster Terzan 5 as part of the Transients and Pulsars with MeerKAT (TRAPUM) Large Survey Project. We observed Terzan 5 atL-band (856–1712 MHz) with the MeerKAT radio telescope for four hours on two epochs, and performed acceleration searches of 45 out of 288 tied-array beams covering the core of the cluster. We obtained phase-connected timing solutions for all ten discoveries, covering nearly two decades of archival observations from the Green Bank Telescope for all but one. Highlights include PSR J1748−2446ao which is an eccentric (e = 0.32) wide-orbit (orbital periodP_b = 57.55 d) system. We were able to measure the rate of advance of periastron (ω̇) for this system allowing us to determine a total mass of 3.17 ± 0.02 M_⊙. With a minimum companion mass (M_c) of ∼0.8 M_⊙, PSR J1748−2446ao is a candidate double neutron star (DNS) system. If confirmed to be a DNS, it would be the fastest spinning pulsar (P = 2.27 ms) and the longest orbital period measured for any known DNS system. PSR J1748−2446ap has the second highest eccentricity for any recycled pulsar (e ∼ 0.905) and for this system we can measure the total mass (1.997 ± 0.006 M_⊙) and estimate the pulsar and companion masses, (1.700_−0.045^+0.015 M_⊙and 0.294_−0.014^+0.046 M_⊙, respectively). PSR J1748−2446ar is an eclipsing redback (minimumM_c ∼ 0.34 M_⊙) system whose properties confirm it to be the counterpart to a previously published source identified in radio and X-ray imaging. We were also able to detectω̇for PSR J1748−2446au leading to a total mass estimate of 1.82 ± 0.07 M_⊙and indicating that the system is likely the result of Case A Roche lobe overflow. With these discoveries, the total number of confirmed pulsars in Terzan 5 is 49, the highest for any globular cluster so far. These discoveries further enhance the rich set of pulsars known in Terzan 5 and provide scope for a deeper understanding of binary stellar evolution, cluster dynamics and ensemble population studies.