A Ricci solition is a Riemannian metric together with a vector field \((M,g,X)\) that satisfies \(\text{Ric}+\tfrac 12 L_Xg=\lambda g\). It is called shrinking when \(\lambda=0\), steady when \(\lambda=0\) and expanding when \(\lambda<0\). If \(X=\nabla f\) the equation becomes \(\text{Ric}+\text{Hess}\,f=\lambda f\) and is called a gradient Ricci solition. Let \(N\) be an Einstein space with Einstein constant \(\lambda\) and let \(\mathbb{R}^k\) act on \(N\) by orthogonal transformations. Then one says that a gradient solition is rigid if it is of the quotient type \(N \times_\Gamma\mathbb{R}^k\). Note that the type of this kind is a flat vector bundle over a base that is Einstein and with \(f=\tfrac 12 d^2\) where \(d\) is the distance on the flat fibres to the base. It is known that not all gradient solitions are rigid, and this paper offers several natural conditions on the curvature that characterize rigid gradient solitions. They are summarized as follows: A shrinking compact gradient solition is rigid with trivial \(f\) if \(\int_M\text{Ric}(\nabla f,\nabla f)\leq 0\). For the expanding and shrinking case a gradient solition is rigid if and only if it has constant scalar curvature and is radically flat, that is, \(\text{sec}(E,\nabla f)=0\), where \(E\) is a curvature basis for \(\nabla f\), that is, \(R(E,\nabla f)\nabla f=\nabla^2_{E,\nabla f} \nabla f-\nabla^2_{\nabla f,E}\nabla f\). Further, all complete non-compact shrinking gradient solitions of cohomology 1 with nonnegative Ricci curvature and \(\text{sec}(E,\nabla F)\geq 0\) are rigid.