In this paper we study the approximability of the Maximum Happy Set problem (MaxHS) and the computational complexity of MaxHS on graph classes: For an undirected graph and a subset of vertices, a vertex v is happy if v and all its neighbors are in S; otherwise unhappy. Given an undirected graph and an integer k, the goal of MaxHS is to find a subset of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a-approximation algorithm for MaxHS on graphs with maximum degree. Next, we show that the approximation ratio can be improved to if the input is a connected graph and its maximum degree is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to proper interval graphs, or block graphs. We prove nevertheless that MaxHS remains NP-hard even for bipartite graphs or for cubic graphs.