  Understanding the material parameters that control the superconducting
transition temperature $T_c$ is a problem of fundamental importance. In many
novel superconductors, phase fluctuations determine $T_c$, rather than the
collapse of the pairing amplitude. We derive rigorous upper bounds on the
superfluid phase stiffness for multi-band systems, valid in any dimension. This
in turn leads to an upper bound on $T_c$ in two dimensions (2D), which holds
irrespective of pairing mechanism, interaction strength, or order-parameter
symmetry. Our bound is particularly useful for the strongly correlated regime
of low-density and narrow-band systems, where mean field theory fails. For a
simple parabolic band in 2D with Fermi energy $E_F$, we find that $k_BT_c \leq
E_F/8$, an exact result that has direct implications for the 2D BCS-BEC
crossover in ultra-cold Fermi gases. Applying our multi-band bound to
magic-angle twisted bilayer graphene (MA-TBG), we find that band structure
results constrain the maximum $T_c$ to be close to the experimentally observed
value. Finally, we discuss the question of deriving rigorous upper bounds on
$T_c$ in 3D.
