Summary: If we have a flow \((X,\mathbb{Z}^m)\) and a cocyle \(h\) on this flow, \(h:X \times \mathbb{Z}^m \to \mathbb{R}^m\), then \(h\) is called close to linear if \(h\) can be written as the direct sum of a linear (constant) cocycle and a cocycle in the closure of the coboundaries. Many of the desirable consequences of linearity hold for such cocycles and, in fact, a close to linear cocycle is cohomologous to a cocycle which is norm close to a linear one. Furthermore in the uniquely ergodic case all cocycles are close to linear. We also establish that a close to linear cocycle which is covering is cohomologous to one with the special property that it can be extended by piecewise linearity to an invertible cocycle from \(X\times \mathbb{R}^m\) to itself. This implies that a suspension obtained from a close to linear cocycle is isomorphic to a time change of the suspension obtained from the identity cocycle.