Known results on cyclic difference sets Is there any infinite family of $v$ for which all the $(v,k,\lambda)$-cyclic difference sets with $k-\lambda$ a prime power coprime to $v$ have been determined? 
A subset $D=\{a_1,\ldots,a_k\}$ of $\mathbb{Z}/v\mathbb{Z}$ is said to be a $(v,k,\lambda)$-cycic difference set if for each nonzero $b\in\mathbb{Z}/v\mathbb{Z}$, there are exactly $\lambda$ ordered pairs $(a_s,a_t)\in D^2$ such that $a_s-a_t=b$. For a $(v,k,\lambda)$-difference set $D$, $k-\lambda$ is called the order.
Let $C_{v,n}$ be the set consisting of all cyclic difference sets of $\mathbb{Z}/v\mathbb{Z}$ with order $n$, and 
$$
C_v=\bigcup\limits_{\text{$n>1$ is a prime power coprime to $v$}}C_{v,n}.
$$ 
For a fixed $v$, $C_v$ can be explicitly written down if $v$ is not too large. My question actually is: have we already know $C_v$ for infinitely many $v$s'? 
I would pose another question related to this: does there exist an $N$ such that for all $v>N$, $|C_v|>0$?    
 A: I'm not sure exactly what you mean because if you prove that there exists a cylcic $(v, k, \lambda)$-difference set for all $v$ except those that are excluded by known nonexistence results, you actually solved the existence problem entirely. So you should impose some condition(s) on parameters. But it can easily become trivial (e.g., cyclic difference sets with the classical parameters). So, it requires some careful and sensible choice of additional condition for your question to make sense and become interesting.
Whatever restriction you choose, here's a recent survey:
Q. Xiang, Recent progress in algebraic design theory, Finite Fields Appl. 11, (2005)
622–653.
And its preprint is available for free from the author's website:
http://titan.math.udel.edu/~xiang/surveyffa.pdf
There have been progress both on their existence and nonexistence. For example, as Theorem 3.4 of the paper says, fairly recently two of the major nonexistence conjectures were proved under the condition that $k-\lambda$ is a prime power greater than $3$. But if you ask if a cyclic $(v,k,\lambda)$-difference set exists for all $v$ that is not excluded by this recent nonexistence result (and the trivial conditions such as meeting the Bruck–Ryser–Chowla condition), then the answer is no, and there are open cases.
Small open parameters can be found in the following table:
http://www.ccrwest.org/diffsets.html
In general, we still have many major open problems when it comes to difference sets. The prime power conjecture (i.e., if $\lambda = 1$, then $n$ is a prime power) is open, too.
