  We discuss the recent developments of projection-based model order reduction
(MOR) techniques targeting Hamiltonian problems. Hamilton's principle
completely characterizes many high-dimensional models in mathematical physics,
resulting in rich geometric structures, with examples in fluid dynamics,
quantum mechanics, optical systems, and epidemiological models. MOR reduces the
computational burden associated with the approximation of complex systems by
introducing low-dimensional surrogate models, enabling efficient multi-query
numerical simulations. However, standard reduction approaches do not guarantee
the conservation of the delicate dynamics of Hamiltonian problems, resulting in
reduced models plagued by instability or accuracy loss over time. By
approaching the reduction process from the geometric perspective of symplectic
manifolds, the resulting reduced models inherit stability and conservation
properties of the high-dimensional formulations. We first introduce the general
principles of symplectic geometry, including symplectic vector spaces, Darboux'
theorem, and Hamiltonian vector fields. These notions are then used as a
starting point to develop different structure-preserving reduced basis (RB)
algorithms, including SVD-based approaches and greedy techniques. We conclude
the review by addressing the reduction of problems that are not linearly
reducible or in a non-canonical Hamiltonian form.
