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Autumn ... actually from the last week of August until the first week of December is pretty good. Plane tickets are usually the cheapest in September and April. The Spring has been getting shorter so it is hard to time the cherry blossoms and not really worth it. The fall leaves are better than cherry blossoms. The Spring also has a strong wind from the Gobi so it picks up dust in the spring. The summer is usually hot and rainy. The winter is not bad. The snow usually melts in Seoul before it hits the ground. However if you love snow then there are places for skiing in the mountains. Avoiding the end of July and beginning of August would be wise.

Why avoid the end of July and beginning of August?
I am more of a warm weather kind of person, not really too fond of snow 😆

38 Celsius and raining does not feel warm. It's a sauna.

It does sound very hot. It's better when it's just sun 😆