Let $(R, {\frak m})$ be a local Cohen–Macaulay ring and let $I$ be an ${\frak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known invariants of $I$. Special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen–Macaulayness of ${\mathcal F}$.