Compact connected iff semi-simple for Lie Groups? Are compact & connected Lie Groups in correspondence with semi-simple Lie groups? I think there is a condition on the center (discrete?) but I'm not sure.
 A: To amplify Ben's answer, I'd point to an earlier post that has lots more detail:   here.   The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps.  Anyway, it's good to browse older Lie group entries on MO first.
PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question.   Textbook material of this kind calls mainly for references rather than discussion.
A: I would like to add the following: 
I think the source of confusion is the fact that the Killing form is nondegenerate (for semi-simple Lie groups) and negative definite (stronger than non-degenerate) for compact Lie groups with trivial center
$SL(2)$ is semi-simple but not compact.
The torus $S^1$ is compact but not semi-simple (abelian).
Compact groups are reductive and semi-simple only when in the case of trivial center.
A: The answer to your title is "no"; lots of semi-simple Lie groups are not compact (for example, $SL_2(\mathbb{R})$).  You're getting this mixed up with the fact that a complex semi-simple Lie group has a unique compact real form, and that this is a bijection to semi-simple compact Lie groups. (Complex reductive groups are in bijection with general compact Lie groups; this allows torus factors on both sides).
