A weaving \(W\) 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 a realization of \(W\), if its projection into the plane is \(W\) 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 a weaving pattern. A weaving pattern is realizable if at least one element of the equivalence class has a 3-dimensional realization. A weaving (pattern) \(W\) is called perfect, if along each line (line segment) of \(W\), the lines intersecting it are alternately ``above'' and ``below''. We prove that (i) a perfect weaving pattern of \(n\) lines is realizable if and only if \(n \leq 3\), (ii) a perfect \(m\) by \(n\) weaving pattern of line segments (in a grid-like fashion) is realizable if and only if \(\min(m,n) \leq 3\), (iii) if \(n\) is sufficiently large then almost all weaving patterns of \(n\) lines are nonrealizable.