Eze seems full of flowers, fountains, archways and tiny nooks and crannies, all very well preserved (like some of its inhabitants, the bandU2 all have houses in the area and co-own a restaurant in the village). There are also leafy squares and cafe's to while away an hour drinking a glass of the local rose or a glass of pastis, or to relax over lunch, but beware the prices wherever there is a view!
Below the village there is the beach of Eze, but it takes about 45 minutes to walk down the very steep path to the beach, The path is called the Frederick Nietzsche path because he wrote the final part of his "Thus Spoke Zarathustra" whilst in Eze. The path is not conducive to anyone except the very fit, especially if you are considering walking up again which will take you considerably longer than it takes to walk down, in fact you should treat Eze beach as an entirely different location to Eze village as it is 400 metres above sea level!.
There are fantastic gardens open to the public, with a lot of local species and a number of tropical species such as bananas, carob trees, oranges date palms and lemon trees all growing wild, which is a testament to the mild winter climate herewith small paths running around the plot, and all in sight of the wonderful blue sea.
Above the village is a huge nature reserve called the "maison de La Nature" literally the house of nature through which you can walk and again admire the stunning scenery.
Eze village has a railway station that can take you into Nice or on to Cannes to the west or through to Monte Carlo and Italy, as Eze is only about a dozen miles from both Monaco and the Italian border, so a short hop on the train will enable you to collect those smelly Italian cheeses or that special olive oil.
Straying in Exe is possible, but in the big hotels can be a very expensive proposition. Many nowadays look to rent a private villa in the area, because you are almost guaranteed great views and some have swimming pools. These are more prevalent than you might think, and many are available all year around. Talk to one of the several agencies in the area which specialise in rental accommodation as they will be certain to find you something that you like which suits your budget.
Getting to Eze is simple. Nice airport is just 15 minutes drive away and flights go to many destinations. You can reach just about any capital city in Europe or at least eight airports in Britain including all 4 London airports. A car would be useful, but not entirely necessary, as you can get a taxi from the airport and the train can take the strain!