Computer-aided diagnosis (CAD) based on histopathological imaging has progressed rapidly in recent years with the rise of machine learning based methodologies. Traditional approaches consist of training a classification model using features extracted from the images, based on textures or morphological properties. Recently, deep-learning based methods have been applied directly to the raw (unprocessed) data. However, their usability is impacted by the paucity of annotated data in the biomedical sector. In order to leverage the learning capabilities of deep Convolutional Neural Nets (CNNs) within the confines of limited labelled data, in this study we shall investigate the transfer learning approaches that aim to apply the knowledge gained from solving a source (e.g., non-medical) problem, to learn better predictive models for the target (e.g., biomedical) task. As an alternative, we shall further propose a new adaptive and compact CNN based architecture that can be trained from scratch even on scarce and low-resolution data. Moreover, we conduct quantitative comparative evaluations among the traditional methods, transfer learning-based methods and the proposed adaptive approach for the particular task of cancer detection and identification from scarce and low-resolution histology images. Over the largest benchmark dataset formed for this purpose, the proposed adaptive approach achieved a higher cancer detection accuracy with a significant gap, whereas the deep CNNs with transfer learning achieved a superior cancer identification.