Quantification of metabolites from magnetic resonance spectra (MRS) has many applications in medicine and psychology, but remains a challenging task despite considerable research efforts. For example, the neurotransmitter $γ$-aminobutyric acid (GABA), present in very low concentration in vivo, regulates inhibitory neurotransmission in the brain and is involved in several processes outside the brain. Reliable quantification is required to determine its role in various physiological and pathological conditions. We present a novel approach to quantification of metabolites from MRS with convolutional neural networks --- MRSNet. MRSNet is trained to perform the multi-class regression problem of identifying relative metabolite concentrations from given input spectra, focusing specifically on the quantification of GABA, which is particularly difficult to resolve. Typically it can only be detected at all using special editing acquisition sequences such as MEGA-PRESS. A large range of network structures, data representations and automatic processing methods are investigated. Results are benchmarked using experimental datasets from test objects of known composition and compared to state-of-the-art quantification methods: LCModel, jMRUI (AQUES, QUEST), TARQUIN, VeSPA and Gannet. The results show that the overall accuracy and precision of metabolite quantification is improved using convolutional neural networks.