Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup S t = e −tH . We identify an ideal n of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n . The regularity is expressed as L 2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in L p , 1 < p < ∞, of some associated Riesz transform operators. Finally, we show that n is the largest ideal of g for which the regularity results hold.Various algebraic characterizations of n are given. In particular, n = s⊕n where n is the nilradical of g and s is the largest semisimple ideal of g.Additional features of this article include an exposition of the structure theory for G in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix. 2000 Mathematics Subject Classification: primary 22E30; secondary 35B65, 58J35. Theorem 1.1 Let m ∈ N 0 and y 1 , . . . , y m ∈ n and set. . , m}, and put w = j 1 + · · · + j m . Then there exists c > 0 such thatNote that for any y 1 , . . . , y m ∈ n we can always choose j 1 = · · · = j m = 1 and w = m in the above theorem, because n ⊆ q = q N ;1 as subspaces. When w > m the theorem gives a more precise bound.Observe that if G is solvable, that is, g = q, then s = {0} because s is semisimple. Thus n = n in this case. It is quite possible to have s = {0} even when G is not solvable. Nevertheless, the following estimate giving an exponential decrease for s-derivatives is of interest. Theorem 1.2 Let y ∈ s, set Y = dL G (y), and let P, Q ∈ R(G). Then there exist positive constants c, σ, b such that P Y QS t 2→2 ≤ c e −σt and |(P Y QK t )(g)| ≤ c e −σt e −b|g| 2 a /t for all t ≥ 1 and g ∈ G.We remark that the crucial feature of s, used to prove Theorem 1.2, is that the corresponding Lie subgroup G s is compact and normal in G.Combining the above theorems we obtain Corollary 1.3 Let m ∈ N 0 , y 1 , . . . , y m ∈ n and set Y i = dL G (y i ). Choose j 1 , . . . , j m ∈ N such that y i ∈ s ⊕ q N ;j i for all i ∈ {1, . . . , m}, and put w = j 1 + · · · + j m . Then there is c > 0 such thatfor all t ≥ 1 and k ∈ {1, . . . , d }.