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We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups $\mathrm{GA}\left(2\right)=\mathrm{GL}(2,ℝ)⋉{ℝ}^2$ and $\mathrm{GA}\left(3\right)=\mathrm{GL}(3,ℝ)⋉{ℝ}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective...