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The group of symplectic matrices is explicitly parametrized, and this description is applied to solve two types of problems. First, we describe several sets of structured symplectic matrices, i.e., sets of symplectic matrices that simultaneously have another structure. We consider unitary symplectic matrices, positive definite symplectic matrices, entrywise positive symplectic matrices, totally nonnegative symplectic matrices, and symplectic M-matrices. The special properties of the LU factorization of a symplectic matrix play a key role in the parametrization of these sets. The second class of problems we deal with is to describe those matrices that can be certain significant submatrices of a symplectic matrix, and to parametrize the symplectic matrices with a given matrix occurring as a submatrix in a given position. The results included in this work provide concrete tools for constructing symplectic matrices with special structures or particular submatrices that may be used, for instance, to create examples for testing numerical algorithms.