Authors’ abstract: We present efficient algorithms for counting points on a smooth plane quartic curve \(X\) modulo a prime \(p\). We address both the case where \(X\) is defined over \(\mathbb{F}_p\) and the case where \(X\) is defined over \(\mathbb{Q}\) and \(p\) is a prime of good reduction. We consider two approaches for computing \(\#X(\mathbb{F}_p)\), one which runs in \(O(p\log p\log\log p)\) time using \(O(\log p)\) space and one which runs in \(O(p^{1/2}\log^2p)\) time using \(O(p^{1/2}\log p)\) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for \(X/\mathbb{Q}\) that compute \(\#X(\mathbb{F}_p)\) for good primes \(p\le N\) in \(O(N\log^3 N)\) time using \(O(N)\) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of \(\mathbb{P}^1\), which in combination with previous results addresses all curves of genus \(g\le 3\). Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest.