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We present F-PKI, an enhancement to the HTTPS public-key infrastructure (or web PKI) that gives trust flexibility to both clients and domain owners, and enables certification authorities (CAs) to enforce stronger security measures. In today's web PKI, all CAs are equally trusted, and security is defined by the weakest link. We address this problem by introducing trust flexibility in two dimensions: with F-PKI, each domain owner can define a domain policy (specifying, for example, which CAs are authorized to issue certificates for their domain name) and each client can set or choose a validation policy based on trust levels. F-PKI thus supports a property that is sorely needed in today's Internet: trust heterogeneity. Different parties can express different trust preferences while still being able to verify all certificates. In contrast, today's web PKI only allows clients to fully distrust suspicious/misbehaving CAs, which is likely to cause collateral damage in the form of legitimate certificates being rejected. Our contribution is to present a system that is backward compatible, provides sensible security properties to both clients and domain owners, ensures the verifiability of all certificates, and prevents downgrade attacks. Furthermore, F-PKI provides a ground for innovation, as it gives CAs an incentive to deploy new security measures to attract more customers, without having these measures undercut by vulnerable CAs.