Due to their computing power, quantum computers may have the disruptive potential to break various currently used encryption and authentication algorithms within the next 15 to 20 years. Once available, quantum computers would threaten currently used asymmetric algorithms such as RSA and elliptic curve cryptography (ECC). An approach that aims to replace RSA and ECC in next generation security protocols is post-quantum cryptography (PQC). In this work, we show the challenges of implementing PQC on embedded devices and smart cards. One important aspect is the protection of schemes against attacks like power analysis and fault injection and research on this topic is still at a very early stage. Moreover, we describe how existing cryptographic hardware on smart cards or embedded microcontrollers can be used to accelerate post-quantum cryptography.