  Let G be a Lie group modelled on a locally convex space, with Lie algebra g,
and k be a non-negative integer or infinity. We say that G is C^k-semiregular
if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover,
the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called
C^k-regular. For G a C^k-semiregular Lie group and m an order of
differentiability, we show that evol is C^m if and only if Evol is C^m. If evol
is continuous at 0, then evol is continuous. If G is a C^0-semiregular Lie
group, then continuity of evol implies its smoothness (so that G will be
C^0-regular), if smooth homomorphisms from G to C^0-regular Lie groups separate
points on G and g is (e.g.) sequentially complete. Further criteria for
regularity properties are provided, and used to prove regularity for several
important classes of Lie groups. Notably, we find that the Lie group Diff(M) of
smooth diffeomorphisms of a paracompact finite-dimensional smooth manifold M
(which need not be sigma-compact) is C^1-regular. We also provide tools which
enable to show that the Lie group of analytic diffeomorphisms of a compact real
analytic manifold is C^1-regular.
