This paper studies generalized variants of the maximum independent set problem, called the Maximum Distance-d Independent Set problem (MaxDdIS for short). For an integer d ≥ 2, a distance-d independent set of an unweighted graph G = (V, E) is a subset S ⊆ V of vertices such that for any pair of vertices u, v ∈ S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r ≥ 3) and planar graphs, as follows: (1) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(r^d-1)-approximation and O(r^d-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(r^d-2/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.