First, some background information:
The way in which the Creator runs the world is termed in classical Jewish sources as hashgaha (השגחה) which means "supervision." The concept of "supervision" is subdivided into the categories of hashgahah p'rattith (השגחה פרטית), "direct (or, specific) supervision," and hashgahah k'lalith (השגחה כללית), "indirect (or, general) supervision." The former being a direct intervention by HaShem into the events of the world, and the latter being that HaShem has pre-programmed the world - and its various components - to function a certain way and thus they continue in their created path.
It is important to understand that hashgaha p'rattith does not indicate "[something] within God's control/knowledge/domain" and hashgaha k'lalith does not indicate "[something] outside of God's control/knowlege/domain" (has wa-shalom). Rather, the entire world is under HaShem's hashgaha - some things being directly effected and/or managed and others being effected and/or managed indirectly, or it is possible to say "actively" and "passively." However, these are terms used by us in order to facilitate understanding and cannot in their fullness apply to the One Transcendent Creator.
Some common examples of hashgaha k'lalith would be leaves falling from trees, spiders catching flies, bigger fish eating smaller fish, et al. While the Creator is aware of these events and designed the nature inherent in each of them, He does not actively determine which leaf will fall or where it will land, which fly will be caught by which spider, or which fish will be swallowed by another. These are natural events that are a part of the world which He has designed to function in this way. Inherent also in hashgaha k'lalith is the idea of "chance" - which is essentially nature taking its course with the scientific variables being too far beyond conscious human perception for us to know with certainty what will happen next. This concept is expressed by Hazal where it states: "Olam k'minhagho nohegh - the world continues on it's natural course" (b. Avodha Zara 54b).
The cognate to this is when the Creator specifically intervenes in the lives of certain humans, "bending" the course of the natural world to accommodate them in various ways according to His will. The condition for this type of hashgaha is that the individual draw close to HaShem's will in thought, word, and deed. To the extent that a person trusts in and aligns himself with the Creator at any given moment, he is able to "draw" - as it were - the Creator's specific supervising influence into the events of his life. A common example of this is the life of Yosef haSsadiq in the Tora.
The writings on these concepts are vast and intricate, but this should suffice as an introduction which will enable you to understand the following answer.
Now, to answer your question:
Most Rishonim - if not all - limit the scope of hashgaha p'rattith in some way. And according to the majority of those do, the world and the various creatures which it contains, are governed mainly by hashgaha k'lalith and that hashgaha p'rattith is limited in scope to human beings - specifically the righteous among the Jewish nation. This view is expressed by the Rambam (Maimonides, More Nevukhim, III:17-18), Ralbag (Gersonides, Sefer Milhamoth HaShem, IV), Rihal (cf. Sefer Kuzari 1:109), Ramban (Nahmanides, Pirush Al HaTora - Shemoth 13:15), and those Rishonim who are essentially Maimonidean in their philosophical orientation (i.e. RaDaQ, Ibn Tibon, Sforno, Me'iri, et al).
In fact, the Sefer HaHinukh (attributed to the Spanish Rishon, Rav Aharon Lewi HaBartziloni) states this limitation quite succinctly:
שיש כתות בני אדם יחשבו כי השגחת הש״י על כל ענייני העולם בין בעלי חיים או כל שאר הדברים כלומר שלא יתנועע דבר אחד קטן בעולם הזה רק בחפצו ב״ה ובגזרתו עד שיחשבו כי בנפול עלה אחד מן האילן הוא גזר עליו שיפול וא״א שיתאחר או יקדם זמן נפילתה אפי׳ רגע וזה דעת רחוק הרבה מן השכל
Translation: "There are sects of people who think that the hashgaha of HaShem Yithborakh is upon all matters of the world, whether animals or other things, that is to say that not even one small thing in the world moves except by His will, blessed be He, and by His decree, to the point that they think that with regard to the falling of one leaf from a tree that it is [Divinely] decreed upon it that it should fall and that it is impossible that it could have fallen either earlier or later than the [Divinely decreed] time of its falling even by a single moment. Such an understanding is very far from intelligent." (Parashath Tazria, Misswah 169)
The idea commonly espoused in the Jewish world today (expressed above as incorrect by the Sefer HaHinukh), namely that everything in the world and everything which happens in the world is a subject to and governed by hashgaha p'rattith, was virtually unknown until the preaching of the Baal Shem Tov. The adage [apparently] spoken by him varies depending on the source retelling it; at times he said to have attributed hashgaha p'rattith to a turning leaf, a blade of grass in the wind, or to grains of sand falling into a hole. In his estimation, the concept of hashgaha k'lalith either did not exist or was an illusion.
L'aniyuth da`ati - in my humble opinion - such a stance is in stark contradiction to centuries, if not millenia, of consistent Tora teaching on the subject (not to mention that such a view leads inevitably to the attribution of absurdity - has wa-shalom - to the Creator, may He be blessed, and a host of other philosophical errors). Therefore, it is to be rejected (cf. Devarim 12-13).