Generic Secure Multi-Party Computation (MPC) was first introduced in the circuit model, using arithmetic circuits, or Boolean circuits. However, many computations are not naturally, or efficiently, representable as circuits. An easy example of this is a binary search over n items. In the RAM model this requires only Θ(log(n)) RAM accesses, but in a circuit model this would require a circuit of size Ω(n).Distributed Oblivious RAM (DORAM) is a functionality that allows reading and writing to a secret-shared memory at a secret-shared location. This is similar to the primitive of Oblivious RAM, in which a program must hide its virtual memory accesses from an adversary who can see which locations it is accessing in physical memory. Many techniques from ORAMs are applicable to DORAMs as well. The thesis makes three contributions in the area of DORAMs and ORAMs. Firstly, it presents an attack on several prominent ORAM and DORAM protocols, and shows a solution which fixes affected protocols at little extra cost. Secondly, it presents a computationally secure DORAM which requires Θ(log(n)(κ + d)) bits of communication per memory access, with much smaller constants than previous work, where κ is a computational security parameter and d is the bit-length of memory blocks. Finally, it presents a statistically secure DORAM which requires Θ(log(n)/ log(log(n))(log^2(n)+d)) bits of communication per memory access. The latter is an asymptotic improvement over previous work, and is the first statistically secure DORAM to require o(log(n)d) communication for blocks of size O(log^2(n)).