In this paper we prove some properties of the nonabelian cohomology H 1 (A, G) of a group A with coefficients in a connected Lie group G. When A is finite, we show that for every A-submodule K of G which is a maximal compact subgroup of G, the canonical map H1 (A, K) → H 1 (A, G) is bijective. In this case we also show that H 1 (A, G) is always finite. When A = Z and G is compact, we show that for every maximal torus T of the identity component G z 0 of the group of invariants G Z , H 1 (Z,T) - H 1 (Z, G) is surjective if and only if the Z-action on G is 1-semisimple, which is also equivalent to the fact that all fibers of H 1 (Z, T)→ H 1 (Z, G) are finite. When A = Z/nZ, we show that H 1 (Z/nZ, T) → H 1 (Z/nZ, G) is always surjective, where T is a maximal compact torus of the identity component G( Z/nZ of G Z/nZ . When A is cyclic, we also interpret some properties of H 1 (A, G) in terms of twisted conjugate actions of G.