A point, usually in three-space, that both an attracts and a repels, attracting in one dimension and repelling to another.

A point on a surface that is neither a peak nor a valley, but still has a 0 gradient.

A fixed point that has at least one positive eigenvalue and one negative eigenvalue in its linearization. More generally, a fixed point for which there are trajectories that tend to the fixed point in both positive and negative time.

a payoff that is simultaneously a row minimum and a column maximum

For a planar system of first-order, autonomous differential equations a critical point where the eigenvalues of the Jacobian matrix evaluated at the critical point are real and of opposite sign. Thus there exists a stable curve containing the critical point such that solutions on this curve approach the critical point as the independent variable goes to positive infinity, and an unstable curve containing the critical point such that solutions on this curve approach the critical point as the independent variable goes to negative infinity.

a molecular geometry such that slight changes cause both a maximum in one direction and a minimum in the other. Saddle points represent a transition structure connecting two equilibrium structures.

In the most general terms, a saddle point for a smooth function (curve, surface or hypersurface) is a point such that the curve/surface/etc. in the neighborhood of this point lies on different sides of the tangent at this point. In certain contexts the definition may vary.