How long for Brownian motion to "fill-out" a torus in d-dimensions? I've been taken by the concise result1
that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes
to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\pi}$.
My question is:

Q. Is the same situation known for the $d$-dimensional 
  torus, $\mathbb{T}^d$? What is the time it takes for a Brownian-motion particle to visit within $\epsilon \to 0$ of
  every point of $\mathbb{T}^d$?

This is probably known—or known to be unknown—so this is just
a reference request.

1
Dembo, Amir, Yuval Peres, Jay Rosen, and Ofer Zeitouni. "Cover times for Brownian motion and random walks in two dimensions." Annals of Mathematics (2004): 433-464.
Annals link.
 A: It looks like the $d$-dimensional case is easier generally than the $d=2$ case according to the excerpt below from this paper (see page three).

...the two-dimensional model is also more difficult than its
  higher-dimensional counterparts. This is because dimension two is critical for
  the walk, resulting in strong correlations. To illustrate the dimension-based
  comparison, observe that very fine results are available for d ≥ 3, see e.g. 2
  and references therein... In contrast, in two dimensions the first-order asymptotics of the
  cover time was completed only recently, after a series of intermediate steps
  over a decade of efforts.

Looking at Aldous and Fill's book (Corollary 7.24) there is an expression for the cover time for $(\mathbb{Z}/n\mathbb{Z})^d$ for $d\geq 3$, $T\sim R_d n \log n$, where 
$$R_d=\int_{0}^1\dots\int_{0}^1\frac{1}{\frac{1}{d}\sum_{u=1}^d(1-\cos{2\pi x_u})}dx_1\dots dx_d.$$
In the limit as $n\rightarrow \infty$ with appropriate normalization, it appears this constant $R_d$ is the cover time.
2: D. Belius (2013) Gumbel fluctuations for cover times in the discrete
torus. To appear in: Probab. Theory Relat. Fields.
Prelininary arXiv abs.
A: A very general answer, in dimension $d\geq 3$, 
 is in the following paper of Dembo, Peres and Rosen
https://projecteuclid.org/euclid.ejp/1464037588:
for compact $d$-dimensional manifolds,
$$C_\epsilon(M) /\epsilon^{2-d}\to V(M)$$
where $V(M)$ is the volume.
The $d$ dimensional case for $d\geq 3$ is much simpler than $d=2$ because in the latter, you have 
long range (=logarithmic) correlations arising from the recurrence of BM. 
In passing, let me mention that Belius's paper mentioned in Josiah Park's answer gives a more refined answer, namely a limit law for a centered version of
$\sqrt{C_\epsilon}$ (with no scaling), 
for the random walk case. Similar results can be written 
for the manifold case in $d\geq 3$. For the
critical case of $d=2$, higher precision than the law of large numbers is available but as of this writing, the limit law for $\sqrt{C_\epsilon}-E\sqrt{C_\epsilon}$ has not been derived.
