The TLS Internet Standard features a mixed bag of cryptographic algorithms and constructions, letting clients and servers negotiate their use for each run of the handshake. Although many ciphersuites are now well-understood in isolation, their composition remains problematic, and yet it is critical to obtain practical security guarantees for TLS. We experimentally confirm that all mainstream implementations of TLS share key materials between different algorithms, some of them of dubious strength. We outline attacks in their handling of resumption and renegotiation, stressing the need to model multiple related instances of the handshake. We study the provable security of the TLS handshake, as it is implemented and deployed. To capture the details of the standard and its main extensions, we rely on miTLS, a verified reference implementation of the protocol. miTLS inter-operates with mainstream browsers and servers for many protocol versions, configurations, and ciphersuites; and it provides application-level, provable security for some. We propose new agile security definitions and assumptions for the signatures, key encapsulation mechanisms (KEM), and key derivation algorithms used by the TLS handshake. By necessity, our definitions are stronger than those expected with simple modern protocols. To validate our model of key encapsulation, we prove that both RSA and Diffie-Hellman ciphersuites satisfy our definition for the KEM. In particular, we formalize the use of PKCS#1v1.5 encryption in TLS, including recommended countermeasures against Bleichenbacher attacks, and build a 3,000-line EasyCrypt proof of the security of the resulting master secret KEM against replayable chosen-ciphertext attacks under the assumption that ciphertexts are hard to re-randomize. Based on our new agile definitions, we construct a modular proof of security for the miTLS reference implementation of the handshake, including ciphersuite negotiation, key exchange, renegotiation, and resumption, treated as a detailed 3,600-line executable model. We present our main definitions, constructions, and proofs for an abstract model of the protocol, featuring series of related runs of the handshake with different ciphersuites. We also describe its refinement to account for the whole reference implementation, based on automated verification tools.