Symmetric and positive semi-definite. Portfolio variance is w'Σw, so the whole of modern portfolio construction runs through it.
Positive semi-definiteness is a real constraint, not a technicality: it is why three assets cannot all be pairwise correlated at -0.9. An interviewer asking whether a given matrix is a valid correlation matrix is testing exactly this.
The estimation problem. With n assets and T observations, the sample estimate is badly conditioned when T is not much larger than n, and is singular when T < n. Shrinkage toward a structured target is the standard fix.