The exceptional Lie group $\mathrm{G}_2$ occurs as the holonomy group of Riemannian metrics on 7-dimensional manifolds. In these lectures I shall explain the constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy $\mathrm{G}_2$. One method uses resolutions of singularities of appropriately chosen 7-dimensional orbifolds with the help of ALE spaces. Another method uses the gluing of two asymptotically cylindrical pieces and requires a certain matching condition for their `boundaries at infinity'.