Let \(f: X\to C\) be a Weierstraß elliptic surface, that is an elliptic surface with a section and with at most rational double points as singularities, whose fibers over the nonsingular curve C of genus \(g\) are all irreducible. The base field is algebraically closed and has characteristic different from two or three. The j-invariants of the fibers of f define a morphism \(j: C\to {\mathbb{P}}^ 1;\) the purpose of this paper is to describe the versal deformation of X in terms of g, \(\chi\) (\({\mathcal O}_ X)\), and the behaviour of j. If \(g\geq 2\) and \(\chi\) (\({\mathcal O}_ X)>(g-1)/2\), or if j is separable and nonconstant, the base space of the versal deformation of X is nonsingular and irreducible around X, and has dimension \(10\chi\) (\({\mathcal O}_ X)+2g-2\); in most other cases, the space of first order deformations of X has a bigger dimension (for details see theorem 2.1), and the surface is usually obstructed. For generic X, the subset of unobstructed first order deformations can be computed explicitly, and this suffices to determine the irreducible components of the base space of the versal deformation for every X (theorem 5.3). The same arguments also show that every elliptic surface with a section having at most rational double points as singularities, defined over an algebraically closed field of characteristic bigger than three, can be lifted to characteristic zero.