Deformable image registration is one of the fundamental tasks in medical imaging. Classical registration algorithms usually require a high computational cost for iterative optimizations. Although deep-learning-based methods have been developed for fast image registration, it is still challenging to obtain realistic continuous deformations from a moving image to a fixed image with less topological folding problem. To address this, here we present a novel diffusion-model-based image registration method, called DiffuseMorph. DiffuseMorph not only generates synthetic deformed images through reverse diffusion but also allows image registration by deformation fields. Specifically, the deformation fields are generated by the conditional score function of the deformation between the moving and fixed images, so that the registration can be performed from continuous deformation by simply scaling the latent feature of the score. Experimental results on 2D facial and 3D medical image registration tasks demonstrate that our method provides flexible deformations with topology preservation capability.