  We present a theory of competing ferromagnetic and superconducting orders in
twisted double bilayer graphene (TDBG). In our theory, ferromagnetism is
induced by Coulomb repulsion, while superconductivity with intervalley
equal-spin pairing can be mediated by electron-acoustic phonon interactions. We
calculate the transition temperatures for ferromagnetism and superconductivity
as a function of moir\'e band filling factor, and find that superconducting
domes can appear on both the electron and hole sides of the ferromagnetic
insulator at half filling. We show that the ferromagnetic insulating gap has a
dome shape dependence on the layer potential difference, which provides an
explanation to the experimental observation that the ferromagnetic insulator
only develops over a finite range of external displacement field. We also
verify the stability of the half-filled ferromagnetic insulator against two
types of collective excitations, i.e., spin magnons and valley magnons.
