For generating correlated normal random variables from independent normals, I know that you can use Cholesky/SVD.
Is there a general method that applies for other random variables, e.g., uniformly distributed ones?
The above is a bit vague, so for discussion purposes, let's constrain this a bit more. How about, for each pair of the dependent random variables, we want to achieve a predefined correlation coefficient.
In general, generating distributions with given properties is a hard problem. However, if you aim to creat a uniform distribution with a given correlation matrix, you can use Gaussian Copula for that.
In particular, given $u_i\sim U_{[0,1]}$ independently, construct $x_i=\Phi^{-1}(u_i)$, transform $y_i = R^{1/2}_{ij}x_j$, where $R_{ij}$ is a desirable correlation matrix, and finally have $v_i=\Phi(y_i)$. Variables $v_i$ are uniformly distributed with correlation matrix $R$ (since homomorphism of each of the variable doesn't change the correlation).