We propose a non-orthogonal multiple access (NOMA) scheme combined with orthogonal frequency division multiplexing access (OFDMA) for visible light communications (VLC), which offers a high throughput, flexible bandwidth allocation and a higher system capacity for a larger number of users. Bidirectional NOMA-OFDMA VLC is experimentally demonstrated. The effects of power allocation and channel estimation on the bit error rate performance are investigated. The experiment results indicate that accurate channel estimation can eliminate the inter-user interference more effectively. The optimum power allocation ratios for uplink and downlink are both about 0.25 in the case of two users.