In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. Motivated by the computation of homophily in social networks, we consider the algorithmic aspects of the following Maximum Happy Edges ($k$-MHE) problem: given a partially $k$-colored graph $G$ and an integer $\ell$, find an extended full $k$-coloring of $G$ making at least $\ell$ edges happy. When we want to make $\ell$ vertices happy on the same input, the problem is known as Maximum Happy Vertices ($k$-MHV). We perform an extensive study into the complexity of the problems, particularly from a parameterized viewpoint. For every $k \geq 3$, we prove both problems can be solved in time $2^n n^{O(1)}$. Moreover, by combining this result with a linear vertex kernel of size $(k + \ell)$ for $k$-MHE, we show that the edge-variant can be solved in time $2^{\ell} n^{O(1)}$. In contrast, we prove that the vertex-variant remains W[1]-hard for the natural parameter $\ell$. However, the problem does admit a kernel with $O(k^2 \ell^2)$ vertices for the combined parameter $k + \ell$. From a structural perspective, we show both problems are fixed-parameter tractable for treewidth and neighborhood diversity, which can both be seen as sparsity and density measures of a graph. Finally, we extend the known NP-completeness results of the problems by showing they remain hard on bipartite graphs and split graphs. On the positive side, we show the vertex-variant can be solved optimally in polynomial-time for cographs.