With the development of machine learning (ML), expectations for artificial intelligence (AI) technology have been increasing daily. In particular, deep neural networks have shown outstanding performance results in many fields. Many applications are deeply involved in our daily life, such as making significant decisions in application areas based on predictions or classifications, in which a DL model could be relevant. Hence, if a DL model causes mispredictions or misclassifications due to malicious external influences, then it can cause very large difficulties in real life. Moreover, training DL models involve an enormous amount of data and the training data often include sensitive information. Therefore, DL models should not expose the privacy of such data. In this paper, we review the vulnerabilities and the developed defense methods on the security of the models and data privacy under the notion of secure and private AI (SPAI). We also discuss current challenges and open issues.