Currently, the global population continues to rise, but an an ever decreasing rate. Extrapolating current trends, the population will stabilize by about 2050 at roughly double today's population. There will be about 10,000,000,000 people on Earth. Fully half of them are likely to have inadequate water, food, and shelter. But not for long. The pessimist on my left shoulder is yelling "Mother Nature's solution is pandemic & starvation."
Side note: Googling "limits to population growth" yields so much subjective crap with a bias toward vegetarianism, globally enforced birth control, assorted eco-disasters, and the soon-to-be global shortage of fresh water, oil, viable farm land, energy, money, medical care, fish stocks, and etcetera, that any real discourse about such limits is buried so deep it is effectively hidden.
I did find one objective (if optimistic) paper on the Biophysical Limits to Global Food Production. It may have glossed over some side effects and uncertainties about high-yield food production, but the answer seems clear: if our #1 priority is feeding people, we can feed a lot of them, even ten billion with a western-style omnivorous diet. Perhaps three times that if we all turn vegan. However, people may have to move to tracts of land that are not highly suited for farming. Luckily, our planet has that type of living space to spare.
Water is a similar issue: if we capture even half of the world's over-land rain, there is no shortage of fresh water, rather a huge surplus. However, there is a distribution problem in time and space, not easily solved. Likewise, if we simply capture icebergs as they break off into the ocean, there is more than enough fresh water available for a population of tens of billions. Another not-so-simple solution, but logical for coastal areas such as Los Angeles. We may have to stop watering our lawns, however.
Touching on a highly religious issue: there may be an energy problem but only if we continue to shackle nuclear power. By now, nearly everyone agrees that fossil fuels are a short term solution and long term problem/crisis-in-the-making. Note that some forms of solar and geothermal energy production could easily satisfy the world's energy needs in 2050. We just need legislation to stop all the naysayers from blocking our path, and high oil prices to make the energy production profitable. Hey, we're half-way there!
Okay, if we put our populations in deserts, on mountains, and floating on the ocean, there is not a space problem. Our planet has lots of room to house few billion more people. Eventually, we'll learn to inject our garbage and trash deep into the crust. This would allow us to stop the continuing settling of cities like New Orleans and Venice while providing a nice, relatively cheap carbon sink to help the excess CO2 problem. Just blenderize it with some waste water and inject the resulting liquid sludge a mile down with a huge pump, and voila. Two problems solved (or at least postponed) in one step.
But I am an optimist.
Many others think that there are many insurmountable problems, including the ones I've touched on: fresh water, adequate nutrition, room, overcrowding, and power.
Many people place the needs of (insert favorite endangered species here, such as the spotted long-neck salamander) ahead of people, and think that maintaining a pristine pre-industrial planet should be our primary goal. That will work, as long as we don't mind killing 90% of the human population to minimize human impacts on the terrestrial ecosystem.
Fully half of the population has an IQ under 100 (duh), and many of them think we should return to the days of living on farms and growing our own food--an agrarian society, or alternatively, living as hunter-gatherers. They somehow imagine an easy existence without any of the benefits our high-tech civilization has provided. I say to them: learn to read.
Personally, I think they are all optimists. Nature has provided natural caps on overpopulation and over-exploitation of available resources: it's called starvation, disease, and pestilence. I fully expect some type of plague to kill half of the world's population, largely confined to the most overcrowded and resource-limited cities and countries. The rest of us will feel bad about it, but frankly there is little we can do.
Unless we start spending money on appropriate long-term solutions, now. Note that I don't mean sending food and water to support the breeders causing the overpopulation problem, but rather in helping them build the infrastructure to solve their problems locally. The key phrase is long term solutions.