For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections? Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.

Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?

(A transvection is a matrix with $1$ everywhere on the diagonal and exactly one other non-zero entry.)
This is certainly the case if $R$ is a field, or if $R$ is a Euclidean domain, but I'm wondering whether there is a complete answer to the question.
 A: I'm answering my own question based on the excellent reference given by Max and the additional comments of Jim Humphreys. There is nothing new in my answer, but I think it's useful to close the question in this way.
Following Hahn-O'Meara, we write $E_n(R)$ for the subgroup of $SL_n(R)$ generated by transvections (also called elementary matrices).

Theorem [H-O'M, Thm 4.3.9]. Let $R$ be a commutative ring. If $R$ is a Euclidean domain or a semilocal ring, then $SL_n(R) = E_n(R)$ for all $n$;
  If $R$ is a Hasse domain of a global field, then $SL_n(R) = E_n(R)$ for all $n \geq 3$ (and in many cases, but not always, also for $n=2$).

There are some other more general results known based on the so-called stable rank of the ring $R$, but as Jim pointed out, it seems hopeless to find a complete answer to the question.
A: The goal of my answer is only to provide recent references.
I warmly recommend these two bits of T. Y Lam's book [2]:

*

*§I.8, for examples where transvections fail to generate $SL_n(R)$

*the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.

And also B. Magurn's latest article on generalized Euclidean group rings [4].
Update.
Here are newer references focussing on the instances of $SL_2(R) \neq E_2(R)$ for $R$ a quadratic order in a totally imaginary quadratic field. The state of the arts is to be found in [3] and [6], while [5] gives a nice geometric insight on the set $SL_2(R)/E_2(R)$.
An older, but in my humble opinion, important paper is [1], where the structure of $SL_2(R)$ as an amalgamated product with factor $E_2(R)$ is described for $R$ the ring of integers of a totally imaginary quadratic field (with few exceptions), see Theorem 2.4.

[1] C. Frohman and B. Fine, "Some amalgam structure for Bianchi groups", 1988.
[2] T. Lam, "Serre's problem on projective modules", 2006.
[3] B. Nica, "The unreasonable slightness of $E_2(R)$ over imaginary quadratic rings", 2011.
[4] B. Magurn, "On a note from Oliver concerning generalized Euclidean group rings", 2014.
[5] K. Stange, "Visualizing the Arithmetic of Imaginary Quadratic Fields", 2017.
[6] A. Sheydvasser, "A Corrigendum to Unreasonable Slightness", 2017.
A: Further results are known: L. Vaserstein's paper "SL_2 of Dedekind rings of arithmetic type" proves these rings are generalized euclidean when they have a unit of infinite order.  Integral group rings of finite groups are generalized euclidean when the group has no homomorphic image among the generalized quaternion groups of order a multiple of 4, no image among the binary polyhedral groups, and the abelianization of the group has generalized euclidean integral group ring.  The finite abelian G with ZG euclidean include the cyclic groups, and Z/2 x Z/2, by the 1984 paper "Generalized euclidean group rings" by Dennis, Magurn & Vaserstein.  But ZG is not generalized euclidean when SK_1(Z[G/[G,G]]) is non-vanishing, as it is for Z/4 x Z/2 x Z/2, for instance.  So this is a delicate property!
A: A nice account of the case
$$
n =2
$$
is given by  I. Reiner in his review of a paper of P.M. Cohn below
The review is very detailed, Hope the tex may compile...
(that NOT worked !)
I give then just the review to try in MR:
MR0207856 (34 #7670)
Cohn, P. M.
On the structure of the ${\rm GL}_{2}$ of a ring.
Inst. Hautes Études Sci. Publ. Math. No. 30 1966 5–53.
20.70 (16.48)
and the beginning of  the review:
This well-written article encompasses a wealth of information about general linear groups over certain classes of rings. 
The author generalizes many earlier results about such groups, and gives a number of new and striking results.
   We proceed to describe some of the main theorems. Assume throughout that the underlying ring $R$ 
has a unity element and is associative, though not necessarily commutative. Denote by $U(R)$ its groups of units.
   (1) Let $\text{GL}_n(R)$ be the group of $n\times n$ invertible matrices over $R$, 
and $D_n(R)$ its subgroup of diagonal matrices. Let $E_n(R)$ be the group generated by the set of transvections
 $\{I+ae_{ij}\colon a\in R,1\leq i,j\leq n,i\neq j\}$, where $\{e_{ij}\}$ is a set of matrix units.
 Define $\text{GE}_n(R)=D_n(R)\cdot E_n(R)$, the subgroup of $\text{GL}_n(R)$ generated by elementary matrices. 
Of course, $E_n(R)\Delta\text{GE}_n(R)$. The author calls $R$ a generalized Euclidean ring (GE-ring) 
if $\text{GL}_n(R)=\text{GE}_n(R)$ for all $n$. 
$$
\dots
$$
