In Metropolis-Hastings Markov chain Monte Carlo, the proposal distribution can be anything including the Gaussian (according to the Wikipedia).
Q: What's the motivation for using anything other than Gaussian? Gaussian works, it's easy to evaluate, it's fast and everybody understands it. Why would I consider anything else?
Q: Since the proposal distribution can be anything, can I use an uniform distribution?
Best Answer
A1: Indeed the Gaussian distribution is probably the most used proposal distribution primarily due to ease of use. However, one might want to use other proposal distributions for the following reason
You can find more discussion here and here.
A2: Yes you can use a Uniform distribution as long as the support for the uniform distribution is bounded (since if the support is unbounded the Uniform distribution is improper as it integrates to $\infty$). So a Uniform on $(x_{t-1} - c, x_{t-1} + c)$.