Higher-dimensional Fáry's theorem? Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges.
For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex embeddable in $\mathbb{R}^d$ has an embedding which is linear on every face?
It is true when $d \ge 2k+1$ by putting things in general position.
I am especially interested to know if anything is known about the case $(k,d)=(2,3)$.
I vaguely remember an old conjecture of Branko Grünbaum that every triangulation of the torus admitting a "straight" embedding in $\mathbb{R}^3$ but I don't know a reference (or whether this is still open).
 A: The answer is negative for all pairs $(k,d)$ with $k+1\leq d\leq 2k$, as long as $k \ge 2$.
Brehm [1] constructed a triangulation of the Möbius strip that does not admit a geometric (simplexwise linear) embedding into $\mathbb{R}^3$.
More generally, for every pair $(k,d)$ with $k+1\leq d\leq 2k$, Brehm and Sarkaria [2]  constructed an example of a $k$-dimensional simplicial complex that admits a piecewise 
linear embedding into $\mathbb{R}^d$, but no geometric embedding.
Moreover, for any given integr $r \geq 0$, there is such a $K$ such that even the $r$-fold barycentric subdivision of $K$ is not geometrically embeddable into $R^d$. 
Furthermore,
for certain values of the parameters, e.g., for $k=d-1$ and $d\geq 5$, it is known that 
there is no recursive bound on the complexity of the subdivision needed to embed a finite
$k$-dimensional simplicial complex piecewise linearly into $\mathbb{R}^d$ (see [3, Corollary 1.2]).
[1] U. Brehm. A nonpolyhedral triangulated Möbius strip. Proc. Amer. Math. Soc., 89(3), 519–522, 1983. 
[2] U. Brehm and K. Sarkaria. Linear vs. piecewise-linear embeddability of simplicial complexes. Tech. Report 92/52, Max-Planck-Institut für Mathematik, Bonn, 1992. Available for
download from Karanbir Sarkaria's website http://kssarkaria.org/List.htm
[3] J. Matousek, M. Tancer, and U. Wagner. Hardness of embedding simplicial complexes in $\mathbb{R}^d$. J. Eur. Math. Soc. 13, 259–295, 2011. 
A: In the paper by Archdeacon et al.,
"Corollary 1.2 proves Grünbaum's conjecture for triangulations of the torus."

Dan Archdeacon, C. Paul Bonnington, Joanna A. Ellis-Monaghan.
  "How to Exhibit Toroidal Maps in Space."
  Discrete & Computational Geometry, 
  Volume 38 Issue 3, October 2007, 
  Pages 573-594.
  (ACM link)

Their introduction says the general conjecture has been disproven:

"When does this embedding have a geometric realization? The problem, restricted to triangulations, was ﬁrst proposed by Grünbaum ([13], Exercise 13.2.3), who conjectured that "Every closed orientable triangulated 2-manifold without boundary can be embedded geometrically in three-dimensional Euclidean space $\mathbb{R}^3$" (see also [6]). This conjecture was recently
  disproven by Bokowski and Guedes de Oliveira [4], who showed that a certain
  triangulation of the complete graph $K_{12}$ on a surface of genus $6$ cannot be
  realized geometrically. Brehm and Schild [5] showed that every triagulation
  of the torus does have a realization in $\mathbb{R}^4$.

