The Harary–Hill Conjecture states that the number of crossings in any drawing of the complete graph $$K_n$$ K n in the plane is at least $$Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor \left\lfloor \frac{n-1}{2}\right\rfloor \left\lfloor \frac{n-2}{2}\right\rfloor \left\lfloor \frac{n-3}{2}\right\rfloor $$ Z ( n ) : = 1 4 n 2 n - 1 2 n - 2 2 n - 3 2 . In this paper, we settle the Harary–Hill conjecture for shellable drawings. We say that a drawing $$D$$ D of $$K_n$$ K n is $$s$$ s -shellable if there exist a subset $$S = \{v_1,v_2,\ldots , v_ s\}$$ S = { v 1 , v 2 , … , v s } of the vertices and a region $$R$$ R of $$D$$ D with the following property: For all $$1 \le i < j \le s$$ 1 ≤ i < j ≤ s , if $$D_{ij}$$ D i j is the drawing obtained from $$D$$ D by removing $$v_1,v_2,\ldots , v_{i-1},v_{j+1},\ldots ,v_{s}$$ v 1 , v 2 , … , v i - 1 , v j + 1 , … , v s , then $$v_i$$ v i and $$v_j$$ v j are on the boundary of the region of $$D_{ij}$$ D i j that contains $$R$$ R . For $$s \ge \lfloor {n/2}\rfloor $$ s ≥ ⌊ n / 2 ⌋ , we prove that the number of crossings of any $$s$$ s -shellable drawing of $$K_n$$ K n is at least the long-conjectured value $$Z(n)$$ Z ( n ) . Furthermore, we prove that all cylindrical, $$x$$ x -bounded, monotone, and 2-page drawings of $$K_n$$ K n are $$s$$ s -shellable for some $$s\ge n/2$$ s ≥ n / 2 and thus they all have at least $$Z(n)$$ Z ( n ) crossings. The techniques developed provide a unified proof of the Harary–Hill conjecture for these classes of drawings.