Let $R$ be a commutative Noetherian ring, $I$ and $J$ be two ideals of $R$, and $M$ be an $R$-module. We study the cofiniteness and finiteness of the local cohomology
module $H^i_{I,J} (M)$ and give some conditions for the finiteness of Hom$_R(R/I, H^s_{ I,J} (M))$ and Ext$^1_R(R/I, H^s_{I,J} (M))$. Also, we get some results on the attached primes of $H^{{\rm dim}M}_{I,J} (M)$.
