The nature of ancient Roman gardens can nowadays be reconstructed (cf Jashemski 1987), and we have ancient literary evidence for the activities which went on in them (cf. Littlewood 1987). As we might expect, the reasons for building some of them were similar to those for erecting great buildings, namely conspicious display. Nero's dual approach of palace and gardens at the Golden House is only the most extravagant of what was a trend (cf. Purcell 1987, especially p.203; he emphasizes pp.190-1 the tyrannical aspects of changing landscape and sea by building hanging gardens and canals).
But if Renaissance gardens can be whimsical, so also can the interpretaions of the antique on which some of them are based. Unfortunately, the renaissance desire to imitate antiquity faced problems, for there was little concrete evidence to draw upon. Actual remains of ancient gardens were scarce and frequently misinterpreted (cf. MacDougall 1972, 40).
What we do know, however, is that several antique gardens too large to be obliterated survived in a ruinous state to be investigated by the Renaissance. Thus the Villa Adriana was a big attraction for the Renaissance, not only to be plundered of its antiquities, but because it represented a superb garden layout as well (cf. Gaffiot & Lavagne for an illustrated survey).The most relevant section is pp.267-371, La fortune de la Villa, which charts its impact from Pirro Ligorio onwards - characteristically, we have no way of tracing its "fortuna" during the Middle Ages, which is a great pity).. Nevertheless, we should be grateful that the villa survived in a state recognisable to be plundered both for its artworks (by Pirro Ligorio), but also for its ideas, as we can easily see by visiting the Villa d'Este and then Hadrian's Villa, its evident source. (cf. Macdonald 1995 on the post-antique impact of this villa).
There were also gardens much closer to Piazza del Popolo, relicts of which probably survived into the Renaissance, although we have no documentation. There were gardens in the Champ de Mars and on the Pincio, with the Horti Lucullani stretching into the current Borghese gardens (cf. Grimal 1984 with its 555-item bibliography. cf. pp.122ff for the gardens of the Champ de Mars and the Pincio; cf. pp.128-130 for the Horti Lucullani, and pp.131-133 for their neighbour to the East, namely the Horti Sallustiani). Just where the Lucullani stopped and the Sallustiani began is in dispute, Grimal complaining (p.129) that Lanciani mixes them up with the Horti Aciliani, auxquels Lanciani attribue les degres et les terrasses dont le magnifique souvenir nous a ete conserve ( cf. his fig.32, with plan according to Ligorio) - an indication that the Pincio part of Valadier's final plan very probably drew inspiration from what he believed to have been there in Antiquity (read G. Kaster, Die Gaerten des Lucullus..., Munich 1974).
If most physical gardens did not survive the Middle Ages (when the large pleasure gardens of the ancients died out or were miniaturised), then descriptions of them certainly did, via literary accounts. One author well known for his love of gardens and the rustic life was the Younger Pliny, whose influence (spliced with that of Vitruvius) begins with the Medici villas of the 15th century, and reaches earnest archaeological reconstructions in the 18th century (cf. du Prey 1994 pp.40-73 for "The Medici and Pliny:". For the archaeological side of the question, cf. Foertsch 1993.
Since one of the main aims of villa life from Antiquity onwards has been to escape from the heat of the town, and since, moreover, Rome offered a plentiful water supply. it is not surprising that the best villas make a feature of water, pools, fountains and even cascades (The Villa d'Este at Tivoli, has them all - of course). Smaller villas even had pools for bathing, such as that in the artificial grotto in Chigi's Villa Suburbana in Trastevere (cf. Charles L. Stinger, Roman Humanist images of Rome, pp.15-38, for Egidio Gallo's poem on the villa; he also cites Blosio Palladio's poem on the same villa, with more wonder at the splendours of the structure and its decoration). As we might expect from the Chigi Chapel in S Maria del Popolo, his villa was intended to evoke wonder for its architecture and decoration (Quinlan-McGrath 1990).
As one might expect, Rome was a focus and an inspiration for the development of grottoes in Antiquity ( cf. Lavagne 1988). CHECK his quotation from Leonardo about grottoes from Cod. Arundel 155r: he doesn't state whether he thinks Leonardo actually saw a grotto (even an ancient one!), or whether this is just Plato's Cave or a variant. Although good on L'Age d'or de la Grotte de Tibere a Hadrien (pp.513-616), he does not concern himself with survival into the Renaissance;
However, the Renaissance certainly knew about ancient fountains from literary descriptions and from survivals, although they apparently had little direct evidence of architectural settings (cf. MacDougall 1978; see pp.93-95). MacDougall makes a distinction between the earlier Renaissance, and the period after 1530s, pointing to fountains in Rome from the 1530s onwards as the break-point, with the earlier ones being of literary inspiration, and the ones after this date to derive from antiquity or what was conceived to have been ancient designs (op.cit., p.101) - although none now remain in their 16th century state. Grottoes are a particular class (op.cit.pp.101-102). For an overview of artificial grottoes, cf. Luchinat 1985.
Given the enthusiasm of the Renaissance for such grottoes - witness the fine one just down the Flaminia in Villa Giulia, recently restored to full beauty, and which has the privilege of being the oldest nymphaeum in Rome, they were clearly conscious of the antique sources of the manner (cf. Rietzsch 1987). More optimistic perhaps than MacDougall, Rietzsch offers several possible antique sources for Renaissance grottoes (see pp.21-29), listing several that the designers of the grotto at the Villa Giulia (and Raphael at Villa Madama) would haved known, such as the Temple of Fortuna Primigenia at Praeneste; the Domus Aurea; and the great Imperial Baths. It would be pleasant to agree, although we simply do not know what was visible at Praeneste (covered by the mediaeval town) or what could have been made out in the burrowing operations into the Domus Aurea. What we can suggest, however, is plenty of other evidence that has simply disappeared in the meantime.
Read with care David Coffin, Gardens and gardening in Papal Rome, pp.285, 193 illus, Princeton 1991;