Deep Learning methods have been gaining a lot of significance for various Biomedical applications for diagnosing several types of diseases. Two applications considered here are: 1) Diabetic Retinopathy Detection and 2) ECG signal Classification (or Arrhythmia Detection). Diabetic Retinopathy (DR) is a major cause of blindness in Diabetic patients, and its early detection benefits diagnosis and subsequent treatment methods. In this work, a Convolutional Neural Network (CNN) using VGG-16 model as a pre-trained neural network has been used for fine-tuning, and, thereby classifying the severity of DR. The model also uses several efficient deep learning techniques including data augmentation, batch normalization, learn-rate scheduling etc., on high resolution images to achieve higher levels of accuracy, greater than previously reported works. For ECG signal classification, CWT (Continuous Wavelet Transforms) were used to convert the time series data into scalogram images. These images were then fed into a Convolutional Neural Network which uses GoogLeNet as a pre-trained network. A classification accuracy of about 90% was obtained. Accuracy levels obtained with GoogLeNet were higher than those obtained using other pre-trained networks such as VGG-16 etc.