This paper characterizes fibrant objects in the model category structure SeCat_c for Reedy fibrant Segal categories. The author proves that Reedy fibrant Segal categories are fibrant in SeCat_c, and that the fibrant objects in SeCat_c are precisely the Reedy fibrant Segal categories. The result is combined with a previous result to show this. The author also shows that the analogous result holds for Segal categories fibrant in the projective model structure on simplicial spaces, considered in the model structure SeCat_f. Segal categories are simplicial spaces that resemble simplicial categories, but their morphisms are only associative up to homotopy. They were introduced by Dwyer, Kan, and Smith. Segal categories can be considered as models for homotopy theories. In any model category, it is important to understand fibrant and cofibrant objects, as they are used to define the homotopy category. In the model structure SeCat_c, all objects are cofibrant, but a characterization of fibrant objects has not been clear. The author provides a complete characterization of fibrant objects in SeCat_c. The author defines Segal precategories and Segal categories, and shows that there are two different model category structures on the category of Segal precategories. In the first structure, SeCat_c, the fibrant objects are Segal categories which are fibrant in the Reedy model category structure on simplicial spaces. The author completes this result by showing that the converse holds, that all Reedy fibrant Segal categories are fibrant in SeCat_c. Similarly, in the second model structure, SeCat_f, the fibrant objects are precisely the Segal categories which are fibrant in the projective model category structure on simplicial spaces. The model category structures on Segal precategories fit into a chain of Quillen equivalences between various model structures. These structures are Quillen equivalent to one another, as well as to a model structure on the category of simplicial categories and to Rezk's complete Segal space model structure on simplicial spaces. The author's interest in comparing these model structures arose from finding models for the homotopy theory of homotopy theories, a project begun by Rezk. The author thanks Bertrand Toën for pointing out an error in a previous proof and the referee for suggestions that led to an improved proof.This paper characterizes fibrant objects in the model category structure SeCat_c for Reedy fibrant Segal categories. The author proves that Reedy fibrant Segal categories are fibrant in SeCat_c, and that the fibrant objects in SeCat_c are precisely the Reedy fibrant Segal categories. The result is combined with a previous result to show this. The author also shows that the analogous result holds for Segal categories fibrant in the projective model structure on simplicial spaces, considered in the model structure SeCat_f. Segal categories are simplicial spaces that resemble simplicial categories, but their morphisms are only associative up to homotopy. They were introduced by Dwyer, Kan, and Smith. Segal categories can be considered as models for homotopy theories. In any model category, it is important to understand fibrant and cofibrant objects, as they are used to define the homotopy category. In the model structure SeCat_c, all objects are cofibrant, but a characterization of fibrant objects has not been clear. The author provides a complete characterization of fibrant objects in SeCat_c. The author defines Segal precategories and Segal categories, and shows that there are two different model category structures on the category of Segal precategories. In the first structure, SeCat_c, the fibrant objects are Segal categories which are fibrant in the Reedy model category structure on simplicial spaces. The author completes this result by showing that the converse holds, that all Reedy fibrant Segal categories are fibrant in SeCat_c. Similarly, in the second model structure, SeCat_f, the fibrant objects are precisely the Segal categories which are fibrant in the projective model category structure on simplicial spaces. The model category structures on Segal precategories fit into a chain of Quillen equivalences between various model structures. These structures are Quillen equivalent to one another, as well as to a model structure on the category of simplicial categories and to Rezk's complete Segal space model structure on simplicial spaces. The author's interest in comparing these model structures arose from finding models for the homotopy theory of homotopy theories, a project begun by Rezk. The author thanks Bertrand Toën for pointing out an error in a previous proof and the referee for suggestions that led to an improved proof.