Optical coherence tomography (OCT) is being increasingly adopted as a label-free and non-invasive technique for biomedical applications such as cancer and ocular disease diagnosis. Diagnostic information for these tissues is manifest in textural and geometric features of the OCT images, which are used by human expertise to interpret and triage. However, it suffers delays due to the long process of the conventional diagnostic procedure and shortage of human expertise. Here, a custom deep learning architecture, LightOCT, is proposed for the classification of OCT images into diagnostically relevant classes. LightOCT is a convolutional neural network with only two convolutional layers and a fully connected layer, but it is shown to provide excellent training and test results for diverse OCT image datasets. We show that LightOCT provides 98.9% accuracy in classifying 44 normal and 44 malignant (invasive ductal carcinoma) breast tissue volumetric OCT images. Also, >96% accuracy in classifying public datasets of ocular OCT images as normal, age-related macular degeneration and diabetic macular edema. Additionally, we show ∼96% test accuracy for classifying retinal images as belonging to choroidal neovascularization, diabetic macular edema, drusen, and normal samples on a large public dataset of more than 100,000 images. The performance of the architecture is compared with transfer learning based deep neural networks. Through this, we show that LightOCT can provide significant diagnostic support for a variety of OCT images with sufficient training and minimal hyper-parameter tuning. The trained LightOCT networks for the three-classification problem will be released online to support transfer learning on other datasets.