OpenAI's Universe utilizes RL algorithms. I also know that Q-learning has been used to solve some games.
Are there any other ML approaches to solve games? For example, could we use genetic algorithms to develop agents that solve games?
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As I see it, it all comes down to game theory, which can be said to form the foundation of successful decision making, and is particularly useful in a context, such as computing, where all parameters can be defined. (Where it runs into trouble is with the aggregate complexity of the parameters per the combinatorial explosion, although Machine Learning has recently been validated as a method of managing intractability specifically in the context of games.)
You might want to check out Playing Games with Genetic Algorithms and Evolutionary game theory.
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Yes, evolutionary algorithms (EAs) can be used to solve/play games too. For example, OpenAI has used evolution strategies (a subset of EAs that uses fixed-length real-valued vectors and self-adaptive mutation rates) to play Atari games. In this blog post, they write
We've discovered that evolution strategies (ES), an optimization technique that's been known for decades, rivals the performance of standard reinforcement learning (RL) techniques on modern RL benchmarks (e.g. Atari/MuJoCo), while overcoming many of RL's inconveniences.
There is also the related paper Evolution Strategies as a Scalable Alternative to Reinforcement Learning (2017) and code.