We identify room-temperature converse magnetoelectric effects (CMEs) that are non-volatile by using a single-crystal substrate of PMN-PT (001)pc (pc denotes pseudocubic) to impart voltage-driven strain to a polycrystalline film of Ni. An appropriate magnetic-field history enhances the magnetoelectric coefficient to a near-record peak of ∼10-6 s m-1 and permits electrically driven magnetization reversal of substantial net magnetization. In zero magnetic field, electrically driven ferroelectric domain switching produces large changes of in-plane magnetization that are non-volatile. Microscopically, these changes are accompanied by the creation and destruction of magnetic stripe domains, implying the electrical control of perpendicular magnetic anisotropy. Moreover, the stripe direction can be rotated by a magnetic field or an electric field, the latter yielding the first example of electrically driven rotatable magnetic anisotropy. The observed CMEs are associated with repeatable ferroelectric domain switching that yields a memory effect. This memory effect is well known for PMN-PT (110)pc but not PMN-PT (001)pc. Given that close control of the applied field is not required as for PMN-PT (110)pc, this memory effect could lead the way to magnetoelectric memories based on PMN-PT (001)pc membranes that switch at low voltage.