A noncollinear resonator with two beams circulating and intersecting in the focus allows resonator-assisted high-harmonic generation of high-power femtosecond pulses with efficient geometrical output coupling. However, the synchronization of the two pulses and alignment of the two beams is a challenge. We describe a resonator design employing wedge mirrors which avoids the need for separate mirrors for the two circulating beams, providing intrinsic synchronization and alignment. Transverse shifting of the wedge mirrors allows fine tuning the relative phase of the pulses, i.e., the interference pattern in the focus. We present the design of two noncollinear resonators at MPQ and at Fraunhofer ILT as well as the results of thorough characterization measurements on a noncollinear resonator with a low-power cw beam, demonstrating its versatility. We also include cylindrical mirrors into the resonator to adjust the ellipticity of the focus, which allows reducing cumulative plasma effects. Ellipticities ɛ>3 are reached without compromising the resonator enhancement of >50. The MPQ resonator has been designed for an EUV frequency comb at 60.8 nm to perform high-precision spectroscopy on the 1⁢𝑠−2⁢𝑠 two-photon transition in H⁡e+ ions for a test of quantum electrodynamics. The ILT resonator has been designed in the framework of developing a VUV frequency comb at around 148 nm envisaged as driver for the low-lying isomeric nuclear transition in 229Th (the so-called thorium isomer) as the presently only candidate for an ultra-high-precision nuclear frequency standard (“nuclear clock”).