The problem of classification of evolution equations \[ (1)\quad v_ t=K(y^ 1,...,y^ n,v,\partial^{i_ 1+...+i_ n}v/\partial y^{i_ 1}...\partial y_ n^{i_ n}), \] where \(y_ 1,...,y_ n\) are space coordinates, with respect to transformations (2) \(t=t(s,x)\), \(y^ j=y^ j(s,x)\), \(v=v(s,x,u)\) is discussed. In general, the right- hand side of the equation obtained by the transformation (2) may contain x explicitly, i.e. loose its evolutionary form (1). This does not occur in the case of trivial transformations (3) \(t=s\), \(y^ j=y^ j(s)\), \(v=v(x,u)\); in this case the transformed equation \[ u_ s=J(x^ 1,...,x^ n,u,\partial^{i_ 1+...+i_ n}u/\partial x^{i_ 1}...\partial x^{i_ n}) \] is called equivalent to (1). It is shown that there is a one-to-one correspondence between classes of equivalent evolution equations related to (1) by transformations of the type (2), nondegenerate in some sense, and point symmetry operators of (1) having the representation \[ Y=\tau (t,y)\partial /\partial t+\sum_{j}\xi_ j(t,y^ 1,...,y^ n)\partial /\partial y^ j+\eta (t,y,v)\partial /\partial v. \] This result is applied to the Hamilton-Jacobi equation for a one-particle Hamiltonian system on a pseudo-Riemannian space with velocity-dependent potential; the Schrödinger equation is also investigated. In the special zero-potential case explicit classification is given.