  We introduce a category of $\mathcal{O}$-orientedsupersingularellipticcurves
and derive properties of the associated oriented and nonoriented $\ell$-isogeny
supersingular isogeny graphs. As an application we introduce an oriented
super-singular isogeny Diffie-Hellman protocol (OSIDH), analogous to the
supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the
commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.
