Three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity $k$ and random additive fitness $\eta$, with rate $(k-1)+ \eta $. For $\eta >0$ we find the connectivity distribution is power law with exponent $\gamma=<\eta>+2$. In the second model (B), the network is built by connecting nodes to nodes of connectivity $k$, random additive fitness $\eta$ and random multiplicative fitness $\zeta$ with rate $\zeta(k-1)+\eta$. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness $(\alpha, \beta)$, $i$ incoming links and $j$ outgoing links gains a new incoming link with rate $\alpha(i+1)$, and a new outgoing link with rate $\beta(j+1)$. The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections.