In the last two decades, the field of DNA-based steganography has emerged as a promising domain to provide security for sensitive information transmitted over an untrusted channel. DNA is strongly nominated by researchers in this field to exceed other data covering mediums like video, image, and text due to its structural characteristics. Features like enormous hiding capacity, high computational power, and the randomness of its building contents, all sustained to prove DNA supremacy. There are mainly three types of DNA-based algorithms. These are insertion, substitution, and complementary rule-based algorithms. In the last few years, a new generation of DNA-based steganography approaches has been proposed by researchers. These modern algorithms overpass the performance of the old ones either by exploiting a biological factor that exists in the DNA itself or by using a suitable technique available in another field of computer science like artificial intelligence, data structure, networking, etc. The main goal of this paper is to thoroughly analyze these modern DNA-based steganography approaches. This will be achieved by explaining their working mechanisms, stating their pros and cons, and proposing suggestions to improve these methods. Additionally, a biological background about DNA structure, the main security parameters, and classical concealing approaches will be illustrated to give a comprehensive picture of the field.