  We employ the nonperturbative functional Renormalization Group to study
models with an O(N_1)+O(N_2) symmetry. Here, different fixed points exist in
three dimensions, corresponding to bicritical and tetracritical behavior
induced by the competition of two order parameters. We discuss the critical
behavior of the symmetry-enhanced isotropic, the decoupled and the biconical
fixed point, and analyze their stability in the N_1, N_2 plane. We study the
fate of non-trivial fixed points during the transition from three to four
dimensions, finding evidence for a triviality problem for coupled two-scalar
models in high-energy physics. We also point out the possibility of
non-canonical critical exponents at semi-Gaussian fixed points and show the
emergence of Goldstone modes from discrete symmetries.
