ASASSN-14ae is a candidate tidal disruption event (TDE) found at the centre of SDSS J110840.11+340552.2 (d ≃ 200 Mpc) by the All-Sky Automated Survey for Supernovae (ASAS-SN). We present ground-based and Swift follow-up photometric and spectroscopic observations of the source, finding that the transient had a peak luminosity of L ≃ 8 × 1043 erg s−1 and a total integrated energy of E ≃ 1.7 × 1050 erg radiated over the ∼5 months of observations presented. The blackbody temperature of the transient remains roughly constant at T ∼ 20 000 K while the luminosity declines by nearly 1.5 orders of magnitude during this time, a drop that is most consistent with an exponential, L ∝ e-t/t 0 with t 0 ≃ 39 d. The source has broad Balmer lines in emission at all epochs as well as a broad He ii feature emerging in later epochs. We compare the colour and spectral evolution to both supernovae and normal AGN to show that ASASSN-14ae does not resemble either type of object and conclude that a TDE is the most likely explanation for our observations. At z = 0.0436, ASASSN-14ae is the lowest-redshift TDE candidate discovered at optical/UV wavelengths to date, and we estimate that ASAS-SN may discover 0.1–3 of these events every year in the future.