In this talk I will present some recent results on dimension of invariant sets, mainly the Julia set and the "conical" Julia set appearing in the dynamics of transcendental maps. I will outline shortly the recent progress in developing the thermodynamic formalism in the context of transcendental dynamics. I will then present the strategy of the proof of Bowen's formula for the dimension of the "conical Julia set". It turns out that the formula can be proved under very mild assumptions about the map in question. Finally, I will speak about the dynamics of the "model" family of transcendental maps $\lambda\exp(z)$. I will formulate several, quite surprising results on the dimension of some invariant sets for this family.