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(rd-1)-approximation and O(rd-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd-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.