For positive integers k and λ, a graph G is (k,λ) -connected if it satisfies the following two conditions: (1) |V(G)|≥k+1, and (2) for any subset S⊆V(G) and any subset L⊆ E(G) with λ|S|+|L| < kλ, G-(S∪L) is connected. For positive integers k and ℓ, a graph G with |V(G)| ≥ k+ℓ+1 is said to be (k,ℓ)-mixed-connected if for any subset S⊆V(G) and any subset L⊆ E(G) with |S|≤ k,|L|≤ℓ and |S| + |L|< k+ℓ, G-(S∪ L) is connected. In this paper, we investigate the (k, λ) -connectivity and (k,ℓ)-mixed-connectivity of random graphs, and generalize the results of Erdős and Rényi (1959), and Stepanov (1970). Furthermore, our argument can show that in the random graph process G~=(Gt)0N, N=(n2), the hitting times of minimum degree at least kλ and of Gt being (k, λ) -connected coincide with high probability, and also the hitting times of minimum degree at least k+ ℓ and of Gt being (k, ℓ)-mixed-connected coincide with high probability. These results are analogous to the work of Bollobás and Thomassen (1986) on classic connectivity.