TLS is a well-known and thoroughly studied security protocol. In this paper, we focus on a specific class of vulnerabilities affecting TLS implementations, state machine errors. These vulnerabilities are caused by differences in interpreting the standard and correspond to deviations from the specifications, e.g. accepting invalid messages, or accepting valid messages out of sequence. We develop a systematic methodology to infer the state machines of major TLS stacks from stimuli and observations, and to study their evolution across revisions. We use the L algorithm to compute state machines corresponding to different execution scenarios. We reproduce several known vulnerabilities (denial of service, authentication bypasses), and uncover new ones. We also show that state machine inference is efficient and practical for integration within a continuous integration pipeline, to help find new vulnerabilities or deviations introduced during development. With our systematic black-box approach, we study over 400 different versions of server and client implementations in various scenarios (protocol version, options). Using the resulting state machines, we propose a robust algorithm to fingerprint TLS stacks. To the best of our knowledge, this is the first application of this approach on such a broad perimeter, in terms of number of TLS stacks, revisions, or execution scenarios studied.