We say that a cryptographic scheme is Continuous Leakage-Resilient (CLR), if it allows users to refresh their secret keys, using only fresh local randomness, such that: • The scheme remains functional after any number of key refreshes, although the public key never changes. Thus, the "outside world" is neither affected by these key refreshes, nor needs to know about their frequency. • The scheme remains secure even if the adversary can continuously leak arbitrary information about the current secret-key, as long as the amount of leaked information is bounded in between any two successive key refreshes. There is no bound on the total amount of information that can be leaked during the lifetime of the system. In this work, we construct a variety of practical CLR schemes, including CLR one-way relations, CLR signatures, CLR identification schemes, and CLR authenticated key agreement protocols. For each of the above, we give general constructions, and then show how to instantiate them efficiently using a well established assumption on bilinear groups, called the K-Linear assumption (for any constant K greater than or equal to 1). Our constructions are highly modular, and we develop many interesting techniques and building-blocks along the way, including: leakage-indistinguishable re-randomizable relations, homomorphic NIZKs, and leakage-of-ciphertext non-malleable encryption schemes.