The maximum entropy principle (MEP) is one of the most prominent methods to investigate and
model complex systems. Despite its popularity, the standard form of the MEP can only generate
Boltzmann-Gibbs distributions, which are ill-suited for many scenarios of interest. As a principled
approach to extend the reach of the MEP, this paper revisits its foundations in information geometry
and shows how the geometry of curved statistical manifolds naturally leads to a generalisation of the
MEP based on the Rényi entropy. By establishing a bridge between non-Euclidean geometry and
the MEP, our proposal sets a solid foundation for the numerous applications of the Rényi entropy,
and enables a range of novel methods for complex systems analysis.