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.