Self-healing materials open new prospects for more sustainable technologies with improved material performance and devices' longevity. We present an overview of the recent developments in the field of intrinsically self-healing polymers, the broad class of materials based mostly on polymers with dynamic covalent and noncovalent bonds. We describe the current models of self-healing mechanisms and discuss several examples of systems with different types of dynamic bonds, from various hydrogen bonds to dynamic covalent bonds. The recent advances indicate that the most intriguing results are obtained on the systems that have combined different types of dynamic bonds. These materials demonstrate high toughness along with a relatively fast self-healing rate. There is a clear trade-off relationship between the rate of self-healing and mechanical modulus of the materials, and we propose design principles of polymers toward surpassing this trade-off. We also discuss various applications of intrinsically self-healing polymers in different technologies and summarize the current challenges in the field. This review intends to provide guidance for the design of intrinsic self-healing polymers with required properties.