The authors study the properties of flexibility and transitivity of automorphism groups for affine algebraic varieties. Given an irreducible affine algebraic variety \(X\), they consider the subgroup \(\mathrm{SAut}(X)\) of \(\mathrm{Aut}(X)\) generated by all one-parameter unipotent subgroups. Denote by \(X_{\mathrm{reg}}\) the smooth locus of \(X\). One says that \(\mathrm{SAut}(X)\) acts infinitely transitively on \(X_{\mathrm{reg}}\) if it acts \(m\)-transitively on \(X_{\mathrm{reg}}\) for all \(m\in{\mathbb{N}}\). A point \(x\in X_{\mathrm{reg}}\) is called flexible if the tangent space \(T_xX\) is spanned by tangent vectors to its orbits under one parameter unipotent subgroups of \(\mathrm{Aut}(X)\). \(X\) is called flexible if all points of \(X_{\mathrm{reg}}\) are flexible. The property of flexibility and its relation to transitivity properties of \(\mathrm{SAut}(X)\) was previously considered in [I. V. Arzhantsev et al., Sb. Math. 203, No. 7, 923–949 (2012); translation from Mat. Sb. 203, No. 7, 3–30 (2012; Zbl 1311.14059)]. In the present article it is proven that, for an affine irreducible variety \(X\) of dimension at least two, the following three conditions are equivalent: (1) \(X\) is flexible; (2) \(\mathrm{SAut}(X)\) acts transitively on \(X_{\mathrm{reg}}\); and (3) \(\mathrm{SAut}(X)\) acts infinitely transitively on \(X_{\mathrm{reg}}\). The authors also study modifications of this result and several applications. In particular, it is shown that affine irreducible affine varieties on which the group \(\mathrm{SAut}(X)\) has an open orbit are exactly those which have a flexible point. They are also characterised as those whose field Makar-Limanov invariant is trivial, and they are always unirational. Finally, the condition of holomorphic flexibility is discussed.