We study the equilibrium orientation of nonspherical Janus particles at an oil-water interface. Two types of nonspherical Janus particles are considered: Janus ellipsoids and Janus dumbbells. To find their equilibrium orientation, we calculate and minimize the attachment energy of each Janus particle as a function of its orientation angle with respect to the oil-water interface. We find that the equilibrium orientation of the interface trapped Janus particles strongly depends on the particle characteristics, such as their size, aspect ratio, and surface properties. In general, nonspherical Janus particles adopt the upright orientation (i.e., the long axis of ellipsoids or dumbbells is perpendicular to the interface) if the difference in the wettability of the two sides is large or if the particle aspect ratio is close to 1. In contrast, Janus particles with a large aspect ratio or a small difference in the wettability of the two regions tend to have a tilted orientation at equilibrium. Moreover, we find that Janus ellipsoids, under appropriate conditions, can be kinetically trapped in a metastable state due to the presence of a secondary energy minimum. In contrast, Janus dumbbells possess only a primary energy minimum, indicating that these particles prefer to be in a single orientation. The absence of a secondary minimum is potentially advantageous for obtaining particle layers at fluid-fluid interfaces with uniform orientation. Our calculation provides a detailed guidance for synthesizing nonspherical Janus particles that can be used as effective solid surfactants for the stabilization of multiphasic fluid mixtures and the modification of the rheological properties of fluid interfaces.