Glioma constitutes 80% of malignant primary brain tumors and is usually classified as HGG and LGG. The LGG tumors are less aggressive, with slower growth rate as compared to HGG, and are responsive to therapy. Tumor biopsy being challenging for brain tumor patients, noninvasive imaging techniques like Magnetic Resonance Imaging (MRI) have been extensively employed in diagnosing brain tumors. Therefore automated systems for the detection and prediction of the grade of tumors based on MRI data becomes necessary for assisting doctors in the framework of augmented intelligence. In this paper, we thoroughly investigate the power of Deep ConvNets for classification of brain tumors using multi-sequence MR images. We propose novel ConvNet models, which are trained from scratch, on MRI patches, slices, and multi-planar volumetric slices. The suitability of transfer learning for the task is next studied by applying two existing ConvNets models (VGGNet and ResNet) trained on ImageNet dataset, through fine-tuning of the last few layers. LOPO testing, and testing on the holdout dataset are used to evaluate the performance of the ConvNets. Results demonstrate that the proposed ConvNets achieve better accuracy in all cases where the model is trained on the multi-planar volumetric dataset. Unlike conventional models, it obtains a testing accuracy of 95% for the low/high grade glioma classification problem. A score of 97% is generated for classification of LGG with/without 1p/19q codeletion, without any additional effort towards extraction and selection of features. We study the properties of self-learned kernels/ filters in different layers, through visualization of the intermediate layer outputs. We also compare the results with that of state-of-the-art methods, demonstrating a maximum improvement of 7% on the grading performance of ConvNets and 9% on the prediction of 1p/19q codeletion status.