Prestressed concrete is a construction technique consisting in applying a bending moment that counteracts the effects of sustained load, thereby reducing the deformability of the structural element. Steel strands are typically used to prestress reinforced concrete (RC) elements. However, exposure to harsh environments may cause premature tendon rupture, eventually leading to structural failure. In this scenario, fiber-reinforced polymer (FRP) tendons offer a potential effective solution due to their fatigue resistance, low weight-to-strength ratio, and corrosion resistance. Despite these advantages, FRP tendons are not commonly used in practice because of the difficulty in anchoring them, which arises from the anisotropic nature and low transverse strength of FRPs. In this paper, two novel mechanical wedge-barrel anchor systems are used to grip FRP bars. The anchor systems differ in geometry and wedge material, namely brass alloy and polymer. Quasi-static tensile tests are performed on glass FRP (GFRP) and carbon FRP (CFRP) bars anchored with the two novel systems. Additionally, preliminary tests with bonded anchors are performed to obtain the mechanical properties of the bars. The experimental results, including tensile strength, failure mode, and anchor efficiencies, are provided and discussed, shedding light on the potential of these new anchor systems depending on the bar type and intended application.