An　orientation　of　a　simple　graph　is　referred　to　as　an　oriented　graph.　Caccetta　and　Häggkvist　conjectured　that　any　digraph　on　n　vertices　with　minimum　outdegree　d　contains　a　directed　cycle　of　length　at　most　⌈n/d⌉.　In　this　paper,　we　consider　short　cycles　in　oriented　graphs　without　directed　triangles.　Suppose　that　α0　is　the　smallest　real　such　that　every　n-vertex　digraph　with　minimum　outdegree　at　least　α0n　contains　a　directed　triangle.　Let　ε　<　(3　−　2α0)/(4　−　2α0)　be　a　positive　real.　We　show　that　if　D　is　an　oriented　graph　without　directed　triangles　and　has　minimum　outdegree　and　minimum　indegree　at　least　(1/(4　−　2α0)+ε)|D|,　then　each　vertex　of　D　is　contained　in　a　directed　cycle　of　length　l　for　each　4　≤　l　<　(4　−　2α0)ε|D|/(3　−　2α0)　+　2.　©　2017,　Institute　of　Mathematics　of　the　Academy　of　Sciences　of　the　Czech　Republic,　Praha,　Czech　Republic.