We construct cubic gravity and its f(P) extension and we investigate their early- and late-time cosmological applications. Cubic gravity is based on a particular invariant P, constructed from cubic contractions of the Riemann tensor, under three requirements: (i) the resulting theory possesses a spectrum identical to that of general relativity, (ii) it is neither topological nor trivial in four dimensions, and (iii) it is defined such that it is independent of the dimensions. Relaxing the last condition and restricting the parameters of cubic gravity we can obtain second-order field equations in a cosmological background. We show that at early times one can obtain inflationary, de Sitter solutions, which are driven by an effective cosmological constant constructed purely from the cubic terms of the simple cubic or f(P) gravity. Concerning late-time evolution, the new terms constitute an effective dark-energy sector and we show that the Universe experiences the usual thermal history and the onset of late-time acceleration. In the case of f(P) gravity, depending on the choice of parameters, we find that the dark-energy equation-of-state parameter can be quintessencelike or phantomlike or it can experience the phantom-divide crossing during the evolution, even if an explicit cosmological constant is absent.