The security of genus $3$ curves in public key cryptography has long been somewhat unclear. For non-hyperelliptic genus $3$ curves Claus Diem found a way to exploit the geometry of the curve to speed up index calculus on the Jacobian, achieving an impressive running time of $\widetilde{O}(q)$. Unfortunately the algorithm suffers from massive memory requirements. We have our own variation of non-hyperelliptic genus $3$ index calculus, which improves Diem’s approach in several ways. We study both the computational complexity and the memory cost of our method in great detail and make the results completely explicit. Combining this with some techniques to alleviate the memory cost, we get a very clear understanding of the security and show that for certain field sizes of practical interest the non-hyperelliptic genus $3$ index calculus is a threat worth taking into account. The so-called isogeny attacks make genus $3$ hyperelliptic curves equally vulnerable.