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We consider a weakly coupled gauge theory where charged particles all havelarge gaps (ie no Higgs condensation to break the gauge "symmetry") and thefield strength fluctuates only weakly. We ask what kind of topological termscan be added to the Lagrangian of such a weakly coupled gauge theory. In thispaper, we systematically construct quantized topological terms which aregeneralization of the Chern-Simons terms and $F\wedge F$ terms, in space-timedimensions $d$ and for any gauge groups (continuous or discrete), using eachelement of the topological cohomology classes $H^{d+1}(BG,\Z)$ on theclassifying space $BG$ of the gauge group $G$.  In 3$d$ or for finite gauge groups above 3$d$, the weakly coupled gaugetheories are gapped. So our results on topological terms can be viewed as asystematic construction of gapped topologically ordered phases of weaklycoupled gauge theories. In other cases, the weakly coupled gauge theories aregapless. So our results can be viewed as an attempt to systematically constructdifferent gapless phases of weakly coupled gauge theories.  Amazingly, the bosonic symmetry protected topological (SPT) phases with afinite on-site symmetry group $G$ are also classified by the same$H^{d+1}(BG,\Z)$. (SPT phases are gapped quantum phases with a symmetry andtrivial topological order.) In this paper, we show an explicit duality relationbetween topological gauge theories with the quantized topological terms and thebosonic SPT phases, for any finite group $G$ and in any dimensions; a resultfirst obtained by Levin and Gu. We also study the relation between topologicallattice gauge theory and the string-net states with non-trivial topologicalorder and no symmetry.

Quantized topological terms in weakly coupled gauge theories and their connection to symmetry protected topological phases