This paper models flexible risers and pipelines as slender elastica structures. The theoretical formulation leads to a type of nonlinear boundary value problem that can be solved numerically given appropriate boundary conditions. The offsetting effects of gravity and buoyancy are included in the analysis. These forces can provide considerable axial loading (as can thermal changes), and hence, stability (buckling) is a major concern. Initial studies are based on the planar problem. A free-vibration analysis is also conducted for small-amplitude oscillations about various deflected equilibrium configurations in terms of natural frequencies and corresponding mode shapes. Energy dissipation and fluid forces are key issues in the forced problem, especially when large deformations are involved. Free vibration information is a vital prerequisite in understanding the response of these types of structures in practice.