A topological space \(X\) is homogeneous if for all \(x,y\in X\) there exists a homeomorphism \(f: X\to X\) such that \(f(x)=y\). In this paper the authors study among other things the class \(\mathcal A\) of compacta that are a continuous image of a homogeneous compactum. It is unknown whether \(\mathcal A\) coincides with the class of all compacta. It is known that all compact metric spaces belong to \(\mathcal A\). Under the Continuum Hypothesis, even all compact first-countable spaces belong to \(\mathcal A\). But so far, no example of a compact homogeneous space with cellularity greater than the continuum is known. The authors obtain several result on (non)homogeneity of products using points of local connectedness (or local contractibility) and components of path connectedness and pose some interesting problems.