After an introduction on classical function spaces, we introduce spaces of Besov and Triebel-Lizorkin type $B^s_{p,q}(\mathbb{R}^n)$ and $F^s_{p,q}(\mathbb{R}^n)$ by Fourier analytical methods and present some properties of those spaces. 

Thereafter, we step up to the scale of function spaces with variable exponents, mainly the variable Lebesgue space $L_{p(\cdot)}(\mathbb{R}^n)$. With this space in mind, we introduce two generalizations of $B^s_{p,q}(\mathbb{R}^n)$ and $F^s_{p,q}(\mathbb{R}^n)$: Besov and Triebel-Lizorkin spaces with variable smoothness and integrability $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$, and 2-microlocal Besov and Triebel-Lizorkin spaces $B^{\bm{w}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\bm{w}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. We focus our attention on the last scale, where some properties will be considered.