Uniform derandomisation of circuit complexity classes
The class uniform-RNC has been studied a lot. 
It is an open problem whether uniform-RNC = uniform-NC. Uniform-(R)NC correspond to (randomized) PRAMs with polynomially many processors and polylogarithmic running time (see the Handbook of Theoretical Computer Science Vol. A). So the question is whether every efficient randomised parallel algorihm can be derandomized.

Since symbolic determinant identity testing is in uniform-RNC, derandomizing RNC implies circuit lower bounds by the results of Kabanets &amp; Impagliazzo (Computational Complexity, 13(1-2), pages 1-46, 2004).

An important special case is the question whether we can compute perfect matchings in uniform-NC. There are several randomized parallel algorithms known, but we do not know whether there is a deterministic one. Recently,
Fenner, Gurjar and Thierauf (STOC 2016) have shown that we can compute perfect matchings in bipartite graphs by uniform circuits of polylogarithmic depth and quasipolynomial size.