Aweaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is above the other. A system of lines (or line segments) in 3-space is called arealization ofW, if its projection into the plane isW and the above-below relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines are combinatorially equivalent and the above-below relations are the same. An equivalence class of weavings is said to be aweaving pattern. A weaving pattern isrealizable if at least one element of the equivalence class has a three-dimensional realization. A weaving (pattern)W is calledperfect if, along each line (line segment) ofW, the lines intersecting it are alternately above and below. We prove that (i) a perfect weaving pattern ofn lines is realizable if and only ifn 3, (ii) a perfect m byn weaving pattern of line segments (in a grid-like fashion) is realizable if and only if min(m, n) 3, (iii) ifn is sufficiently large, then almost all weaving patterns ofn lines are nonrealizable.