Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution? One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is,
Every symmetric positive semi-definite matrix is a covariance marix of some multivariate distribution.
Is it true? If it is, how can we prove it?

Yes.
Let $M$ be symmetric and PSD. Since it is symmetric PSD there is cholesky factorization $LL^T$. Given a random vector $Z$ with covariance $I$ and zero expectation we construct $X = LZ$. Then
$$
V[X] = E[XX^T] - 0 = E[(LZ) (LZ)^T] = E[LZZ^TL^T] = LE[ZZ^T]L^T = LL^T = M.
$$
So $M$ is the covariance matrix of $X$. Thus, every symmetric positive definite matrix is a covariance matrix.

Note that both the symmetric and the PSD property are required for the cholesky decomposition:
If $A$ decomposes to $LL^T$ then
$$
x A x = x LL^T x = (xL, xL) = \lVert xL \rVert^2 \geq 0
$$
so $A$ must be PSD. Further $A^T = (LL^T)^T = LL^T = A$ so $A$ must be symmetric.
This answer assumes real matrices.
