Here’s a topic that needs a simple synopsis.

Assume:
$\beta \sim \mathcal{N}(\beta_0, \Sigma_\beta)$
$y | \beta \sim \mathcal{N}(X\beta, \Sigma_y)$

Then:
$\beta | y \sim \mathcal{N}(\hat{\beta}, V_\beta)$

Where:
$V_\beta = (X^\top\Sigma_y^{-1}X + \Sigma_\beta^{-1})^{-1}$
$\hat{\beta} = V_\beta (X^\top\Sigma_y^{-1}y + \Sigma_\beta^{-1}\beta_0)$

References: here and here