The results from Planck2015, when combined with earlier observations from WMAP, ACT, SPT and other experiments, were the first observations to disfavor the "classic" inflationary paradigm. To satisfy the observational constraints, inflationary theorists have been forced to consider plateau-like inflaton potentials that introduce more parameters and more fine-tuning, problematic initial conditions, multiverse-unpredictability issues, and a new 'unlikeliness problem.' Some propose turning instead to a "postmodern" inflationary paradigm in which the cosmological properties in our observable universe are only locally valid and set randomly, with completely different properties (and perhaps even different physical laws) existing in most regions outside our horizon. By contrast, the new results are consistent with the simplest versions of ekpyrotic cyclic models in which the universe is smoothed and flattened during a period of slow contraction followed by a bounce, and another promising bouncing theory, anamorphic cosmology, has been proposed that can produce distinctive predictions.The Planck2015 [1] and Planck2013 [2] observations, combined with the results by the Wilkinson Microwave Anisotropy Probe (WMAP), Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT) teams have shown that the measured spatial curvature is small; the spectrum of primordial density fluctuations is nearly scale-invariant; there is a small spectral tilt, consistent with a simple dynamical mechanism that caused the smoothing and flattening; and the fluctuations are nearly gaussian. These features are all consistent with the simplest textbook inflationary models. At the same time, Planck2015 also confirmed that r, the ratio of the tensor perturbation amplitude to the scalar perturbation amplitude, is less than 0.1, a result that virtually eliminates all the simplest textbook inflationary models. The development is notable because, as emphasized by Ijjas et al. [3], it is the first time that the classic inflationary picture has been in conflict with observations.The results have led theorists to consider alternative plateau-like inflationary models whose parameters can be adjusted to reduce the expected value of r while retaining all the rest of the classic predictions. However, as we explain in this brief review, the remaining models raise new issues. They require more parameters, more tuning of parameters, more tuning of initial conditions, a worsened multiverse-unpredictability problem, and a new challenge that we call the inflationary 'unlikeliness problem. ' One response to these problems has been that they should be ignored. The classic inflationary picture should be abandoned in favor of an alternative 'postmodern' inflationary picture that allows enough flexibility to fit any combination of observations. The classic and postmodern inflationary pictures are so different that they ought to be viewed as distinct paradigms to be judged separately.A more promising response to Planck2015 has been to develop "bouncing" cosmologie...