A universal central extension of a group G is a central extension, G, of G satisfying the additional conditions (i) G = [G, G], and (ii) every central extension of G is split. If, by analogy with topology, we call a group G connected if H1(G, Z) = 0 and simply connected if H1(G, Z) H(G, Z) = 0, the above conditions are equivalent to saying G is simply connected. In this language, a universal central extension of G is called a universal covering. Now suppose b is a reduced irreducible root system, A is a commutative ring with unit, and (cD, A) is the elementary subgroup of the points in A of a Chevalley-Demazure group scheme with root system (. Define the Steinberg group, St ((, A), by generators and relations as in [19]. Then the main concern of this paper is to determine under what conditions ((, A) is connected and St ((, A) is simply connected. When A is a field, these questions were posed and resolved in a well-known paper of Steinberg [19]. For the groups of type Al, 1 ? 2, they have been treated by Kervaire [10] and Steinberg [20]. The results of this paper were announced in [16]. Connectivity is discussed in Section 4. For groups of rank > 2, the main result is most suggestively stated E ((, A) and St ((F, A)) are connected if and only if ((, A/m) is connected for every maximal ideal m C Thus these groups are connected unless ( C= or G2 and A has a residue field with. two elements. The above statement is also true for groupsof rank 1, provided A is semi-local. In fact, a stronger result holds in the rank 1 case: (A1, A) and St (A,, A) are connected if the ideal generated by {u2 _1, u C A*} is all of A. Although this implies that A has no residue field with 2 or 3 elements, I do not know whether these are, in general, equivalent conditions. Despite the relationship with the case of fields in the formulation. of these theorems, their proofs do not resemble the proofs for fields. They are based instead on careful exploitation of commutator relations in the groups of rank 2. To decide when St ((, A) is simply connected, it remains to determine when every central extension of St ((, A) is split. The answer, roughly