Let \(c:I\to\mathbb{R}^3\), \(s\to c(s)\) be a regular curve of class \(C^3\) with arc length parameter \(s\in I\) and curvature \(\kappa(s)>0\), i.e., a Frenet space curve. Denote by \(\tau\) the torsion of \(c\) and by \(\sigma\) its spherical arc length parameter which is defined – up to a translation – by \(d\sigma/ds= \kappa(s)\). Then the ordered pair \((\kappa^*,\tau^*)\) of functions \(\kappa^* (\sigma):=-\kappa'(\sigma)/ \kappa(\sigma)\) and \(\tau^*(\sigma):=\tau(\sigma)/ \kappa(\sigma)\) is called a (local) shape of \(c\); it consists of the shape curvature \(\kappa^*(\sigma)\) and the shape torsion \(\tau^*(\sigma)\) of the curve \(c\) which form a pair of fundamental invariants under the group of direct similarities of the Euclidean space \(\mathbb{R}^3\) (theorem 1/lemma 1). The authors prove the following analogue of the fundamental theorem of space curves (theorem 3): Let \(f_i:I\to\mathbb{R}\), \(i=1,2\), be two functions of class \(C^1\); modulo a direct similarity of \(\mathbb{R}^3\) there exists a unique Frenet curve with the shape curvature \(f_1\) and the shape torsion \(f_2\). Three examples show which space curves are recovered from special shapes, namely – using real constants \(a\neq 0\) and \(b\neq 0\): (1) a circular helix from the shape \((0,a\)); (2) a conic spiral from the shape \((b,a)\); (3) a (special) cylindrical helix having the shape \((1/\sigma,a)\).