Nowadays, more and more problems are solved through specific manifold formulation. This often allows important reduction of computation time and/or memory management compared to classical formulations on the classical Euclidean space (because of non-linear constraints like restricting the solutions to a certain subdomain of a larger ambiant space). Interpolation and optimization tools can be useful for solving some of these problems (like defining the optimal trajectory of a humanitory plane dropping supplies, or fitting two objects orientations). However, current procedures are only defined on the Euclidean space. In this presentation, I focus on interpolation methods and, more precisely, I propose a new general framework to fit a path through a finite set of data points lying on a Riemannian manifold. The path takes the form of a continuously-differentiable concatenation of Riemannian Bézier segments. This framework will be illustrated by results on the Euclidean space, the sphere, the orthogonal group and the shape manifold.​ The content of this presentation meets also very recent research carried out in this institute for providing novel efficient manifold-based optimization methods.