The authors examine connections between the contact topology and Beltrami fields in hydrodynamics on Riemannian manifolds in dimension 3. The following results are proved: 1) the class of (non-singular) vector fields on a 3-manifold parallel to their (non-singular) curl is identical to the class of Reeb fields under rescaling; 2) every \(C^\omega\) steady solution to Euler equations for perfect incompressible fluid on \(S^3\) possesses a closed flowline; 3) any \(C^\infty\) steady rotational Beltrami field on \(T^3\) which is homotopically non-trivial must have a contractable closed flowline, and 4) any \(C^\omega\) steady Euler flow on \(T^3\) which is homotopically non-trivial must have a closed flowline.