  Let $A$ be a non-zero positive bounded linear operator on a complex Hilbert
space $(\mathcal{H},\langle\cdot,\cdot\rangle)$. Let $\omega_A(T)$ denote the
$A$-numerical radius of an operator $T$ acting on the semi-Hilbert space
$(\mathcal{H},\langle\cdot,\cdot\rangle_A)$, where $\langle x, y\rangle_{A}
:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. Let $N_A(\cdot)$ be a
seminorm on the algebra of all $A$-bounded operators acting on $\mathcal{H}$
and let $T$ be an operator which admits $A$-adjoint. Then, we define the
generalized $A$-numerical radius as
$$\omega_{N_A}(T)=\displaystyle{\sup_{\theta \in \mathbb{R}}}\;
N_A\left(\frac{e^{i\theta}T+e^{-i\theta}T^{\sharp_A}}{2}\right),$$ where
$T^{\sharp_A}$ denotes a distinguished $A$-adjoint of $T$. We develop several
generalized $A$-numerical radius inequalities from which follows the existing
numerical radius and $A$-numerical radius inequalities. We also obtain bounds
for generalized $A$-numerical radius of sum and product of operators. Finally,
we study $\omega_{N_A}(\cdot)$ in the setting of two particular seminorms
$N_A(\cdot)$.
