The purpose of this dissertation is to describe in detail the algebra of central measures ZM(G) for a connected Lie group G, with no compact connected normal semisimple subgroups. This together with results of Ragozin ({Ra,1} and {Ra,2}) yields a description for the algebra ZN(G) for a connected Lie group G We study the closures of conjugacy classes which are relatively compact. Along the way we study the extreme central probability measures and get a one-to-one correspondence between the extreme central probability measures and the closures of conjugacy classes which are relatively compact. This correspondence is given such that, each extreme central probability measure is supported on a such conjugacy class (which is compact and unique); also for each conjugacy class, with compact closure, there exists only one extreme central probability measure such that the closure of this class is its support We show also that under certain restriction on (mu) (ELEM) ZM(G), (mu)('*n) is in a subalgebra of a simple type, and every h (ELEM) (DELTA)ZM(G) is determined by its value on such subalgebras. These simple subalgebras are nearly convolution measure algebras in the sense of Taylor {Ta}. So the study of (DELTA)ZM(G) becomes a study of objects like convolution measure algebras To describe the subalgebras mentioned above, we include, as tools, some properties of a locally compact abelian group and its closed analytic subgroups. In fact, we study for each closed analytic subgroup H, an L-ideal M(S,H) in M(S) consisting of measures absolutely continuous on H cosets