This paper presents a new method, named implicit and explicit feature purification (IEFP), for age-invariant face recognition. Facial features extracted from a face image contain the information about the identity, age, and other attributes. For age-invariant face recognition, it is important to remove the irrelevant information, and retain the identity information only, in the facial features. Through the two proposed feature purification mechanisms, our framework can produce facial-feature embeddings that preserve identity information as much as possible and are insensitive to age variations. Specifically, on the one hand, a special network module is devised to implicitly purify the original facial features obtained from a face encoder. On the other hand, to obtain purer facial feature representations for age-invariant face recognition, irrelevant information within the implicitly purified features, such as the age, is further removed. This is realized by using a regularizer, based on information theory, to explicitly minimize the correlation between identity-related features and age-related features. Comprehensive ablation studies show that these two feature purification schemes can work independently, as well as collaboratively, to achieve better performance. Extensive evaluations on several benchmark data sets show that the IEFP method is on par with those competitors learned on far more favorable training samples, and it achieves the best performance in a fair comparison. Furthermore, we provide mathematical interpretation to explain the effectiveness of our approach, and find that it tends to generate low-rank, yet high-dimensional, representations for age-invariant face recognition.