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In the present paper, the motion of finite-core vortices in the presence of both infinite and semi-infinite bodies is considered. The vortices are massive in the sense that the cores are assumed to be inviscid with constant vorticity after Batchelor [J. Fluid Mech. 1, 177 (1956)]; this requires that the core radius is much greater than the inverse square root of the appropriate Reynolds number. Four geometries are considered: the plane wall, a corner, a thin semi-infinite plate, and a semi-infinite wedge. The flow field is assumed to be inviscid, two-dimensional, and irrotational outside the core of the vortex and the shapes of the vortices are computed as solutions to an initial value problem. Results for the plane wall and the corner flow suggest that the vortices do not rebound due to the effect of the finite core; moreover, the motion of the centroid of the vortex is virtually identical to the motion of the associated point vortex of the same strength. The computed shapes of the vortices are, in general, not elliptical. For the semi-infinite plate and wedge, violent collisions with the leading edge may occur depending on the initial position of the vortex. The results for the wedge are compared with the flow visualization studies of Ziada and Rockwell [J. Fluid Mech. 118, 79 (1982)].