We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general q2 are expressed in terms of multiple polylogarithms, and results for fiftyone master integrals at the threshold q2 = 4m2 are expressed in terms of multiple polylogarithms of argument one, with indices equal to zero or to a sixth root of unity
